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TEACHERS COLLEGE, COLUMBIA UNIVERSITY 

CONTRIBUTIONS TO EDUCATION 

NO. 8 



THE EDUCATIONAL SIGNIFICANCE 

OF SIXTEENTH CENTURY 

ARITHMETIC 

FROM THE POINT OF VIEW OF THE PRESENT TIME 



BY 

LAMBERT LINCOLN JACKSON, Ph. D. 

Head of the Department of Mathematics 
State Normal School, Brockport, N. Y. 




PUBLISHED BY 

Heacbers College, Columbia TUntversitr 

NEW YORK 
1906 



[USRARY of CONGRESS I 

Two Copies Received 

JAN 21 130? 

u CopyriffM Entry 

}Wa,. v <f . ffC 

CLASS A XXc„No. 

bOPY'B. 






PREFACE 



Although there is much material on the history of six- 
teenth century arithmetic, comparatively little has been written 
on the teaching of the subject at that time, and less upon the 
educational significance of that teaching from the point of 
view of the present. 

This dissertation is the result of a research into the arith- 
metic of the fifteenth and sixteenth centuries for the purpose 
of showing its bearing upon the present teaching of the sub- 
ject. The exact period chosen is that from 1478, the date of 
the first printed arithmetic, to 1600, because between these 
dates arithmetic took on the form which it retained for three 
centuries, besides setting forth for the first time its educational 
functions. So little was written at that time on the educational 
values of studies that it is necessary to examine with unusual 
care the treatises and text-books on arithmetic in order to de- 
termine the principles and aims which guided their authors 
in creating them. 

No general history of mathematics can be of use in a re- 
search of this kind, unless it contains extensive extracts from 
the original sources, and there is only one such work, namely, 
that of Cantor. 1 But, if one were to depend upon Cantor for 
a treatment of sixteenth century arithmetic, he would obtain 
an exhaustive knowledge of the leading writers only, whereas 
the great mass of minor works is necessary to supply a large 
portion of the data needed to determine the educational signi- 
ficance of the subject. No one has expressed more clearly 
than DeMorgan 2 the relative value of major and minor works 
for the purpose of interpreting history. " Unfortunately, his- 
tory must of necessity be written mostly upon those works 

1 Cantor, M., " Vorlesungen tiber Geschichte der Mathematik," 3 vol. 
(Leipsic, 3d ed., 1900). 

2 DeMorgan, A., "Arithmetical Books" p. vi (London, 1847). 

3 



4 PREFACE 

which, by being in advance of their age, have therefore be- 
come well known. It ought to be otherwise, but it cannot be, 
without better preservation and classification of the minor 
works which people actually use, and from which the great 
mass of those who study take their habits and opinions ; — or — 
until the historian has at his command a readier access to 
second and third rate works in large numbers ; so that he may 
write upon the effects as well as the causes." Other general 
histories of mathematics like those of Kastner, 1 Suter, 2 
Hankel, 3 Gunther, 4 and Gerhardt 5 contain very brief exposi- 
tions of the arithmetic of the Renaissance, and all having been 
written before Cantor's work furnish little, if anything, of 
importance not covered by him. 

Among those histories devoted entirely to arithmetic and 
drawn from original sources, only two discuss to any consider- 
able extent the works which form the basis of this disserta- 
tion. These are Unger's, 6 chiefly on the arithmetic of the 
Germans, and Peacock's 7 on the histoiry of arithmetic in gen- 
eral. Unger's treatise, because of its accuracy and its sys- 
tematic form, has become the standard work on the history 
of arithmetic and its teaching in Germany. Its foundation, 
however, is limited, for, although thirty-two original works 
oif the period in question are mentioned, only twenty are 
drawn upon, three being Italian, one French, one Flemish, and 
fifteen German. Thus, Unger's work, excellent as it is, does 
not furnish the historical material necessary for a study of six- 
teenth century arithmetic from an international point of view. 
Peacock's article is the most extensive exposition of the sub- 

1 Kastner, "Gesdiichte der Mathematik" (1796-1800). 

2 Suter, "Gesdiichte der math. Wissenschaften " (1873-75). 

3 Hankel, "Zur Gesdiichte d. Math, in Alterthum a. Mittelalter" (1874). 

4 Gunther, " Vermischte Unitersuchungen z. Geschichte d. math. Wissen- 
schaften" (1876). 

5 Gerhardt, "Gesohichte der Mathematik in Deutschland " (1877). 

6 Unger, " Die Methoidik der praktischen Arithmetik in historischer Ent- 
wickelung vom Ausgange des Mittelalters his auf die Gegenwart" (1888). 

7 Peacock, " History of Arithmetic " in Encyclopedia Metropolitana, vol. 
1, pp. 369-476 (London, 1829). 



PREFACE 5 

jeet, but, besides emphasizing the Italian works in undue 
proportion, it omits an important link in the French contribu- 
tion and ignores the Dutch arithmetic altogether. Had Pea- 
cock been able to supply these important departments, and 
had he, like Unger, added a history of the teaching of the sub- 
ject, the historical research on' which the present dissertation 
is based would have been unnecessary. Besides the histories 
of Unger and Peacock there are several important works 
which contain matter germane to' this subject, but they deal 
either with a small portion of the history of the period in 
question or with some particular phase of it. Thus : Giinther, x 
who probably is second only to' Cantor as an authority on the 
early history of arithmetic, closed his work with the critical 
date, 1525. Sterner 2 covers practically the same ground as 
Unger, but draws the bulk of his illustrative material from 
the arithmetics of Kobel, 3 Boschenstein, and Riese only, and 
gives a much inferior treatment of the teaching of the sub- 
ject. Heinrich Stoy 4 gives very little concerning six- 
teenth century arithmetic and confines his investigation to' the 
development of the number concept and to* the various modes 
of its expression. Grosse, 5 as indicated by the title of his 
book, discusses those arithmetics which apply the rules and 
processes to quantitative data which have a historical setting; 
for example, the periods of reigning dynasties or the size of 
armies that participated in the great battles of the past. There 
were only a half-dozen such arithmetics of which Suevus's 
and Meichsner's are the best types. Treutlein, 6 Villicus, 7 and 

1 Giinther, " Geschichte des math. Unterrichts im deutsichen Mittelalter 
bis 1525 (in Monumenta Germaniae Paedagogica, 1887). 
bis 1525 (in Monumenta Germaniae Paedagogica, 1887). 

3 See bibliographical list, page 1 of Sterner' s work. 

4 Stoy, H., "Zur Geschichte des Rechenunterrichts " (Diss., 1876). 

5 Grosse, "Historische Rechenbucher des 16 und 17 Jahrhunderts (1901). 

6 Treutlein, "Das Rechnen im 16. Jahrhundert " (1877). 

7 Villicus, " Das Zahlenwesen der Volker im Altherthum u. die Enrwick- 
elung d. Zifferrechnens " (1880). 

"Geschichte der Rechenkunst" (a slight revision of the former work) 
(1897). 



6 PREFACE 

Kuckuck * are other German writers who touch this period, 
but their works are earlier and briefer than that of Unger. 
Tireutlein's work is a standard, but it treats of a limited num- 
ber of authors, mostly German, emphasizes a few special sub- 
jects, and neglects close comparative study. Leslie 2 deals 
with the theory of calculation in its historical development. 
For example, he pays much attention to the reasons for the 
origin of the various scales of notation, and to the systems of 
objective representation of numbers, called Palpable arithme- 
tic. The subject-matter of sixteenth century arithmetic found 
in this work is very limited. It is unnecessary to mention 
more recent philosophical treatises, like Brooks's, 5 because they 
contain no history of value, with which this article is con- 
cerned, not found in Peacock or Leslie. 

Thus, in order to accomplish the purpose of this disserta- 
tion, it was necessary to make a more extended and systema- 
tic research into the arithmetic of the fifteenth and sixteenth 
centuries than has hitherto been made. It was imperative 
first of all to supply those departments of French and Dutch 
arithmetic missing from the best historical treatises. Then, 
it was necessary to consult the original sources already ex- 
amined by others in order to find matter of educational signi- 
ficance, much of which had not been* noted by other inves- 
tigators, as well as to obtain a broader basis for detailed 
comparisons. 

The dissertation is divided into two chapters. Chapter I 
contains the result of the research into the subject-matter 
and teaching of arithmetic in the fifteenth and sixteenth cen- 
turies. Chapter II contains an exposition of the bearing of 
the arithmetic of that period upon the present teaching of 
the subject. 

It is a great satisfaction to the author to acknowledge his 
indebtedness to Professor David Eugene Smith of Teachers 

1 Kuckuck, "Die Rechenkunst im 16. Jahrhundert " (1874). 

2 Leslie, "Philosophy of Arithmetic," Edinburgh (1820). 

3 Brooks, "Philosophy of Arithmetic" (1901). 



PREFACE 7 

College, Columbia University, for directing" the research, to 
George A. Plimpton, Esq., of New York City, for access to> his 
extensive collection of rare mathematical works, a privilege 
which alone made this research unique, and to Professor 
Frank M. McMurry of Teachers College, Columbia Univers- 
ity, for his help in making the article of practical value. 

L. L. Jackson. 

Brockport, New York, 1905. 



BIBLIOGRAPHY 



Original Sources on Which This Research is Based 

Anonymous, " Thaumaturgus Mathematicus, Id est Admirabilium effec- 
torum e mathematicarum disciplinarum Fontibus Profluentium Sylloge." 

Munich, 165 1. 

Anonymous. Treviso Book. (So called from the place of printing. 

This is the earliest printed Arithmetic known to exist.) " Incommincio 

vna practica molto bona et vtile a eiafchaduno chi vuole vxare larte dela 

merchadantia chiamata vulgarmente larte de labbacho." 
Colophon: "A Triuifo :: A di. 10. Deceb 4 :: .1478." 
It contains 123 pages, not numbered. Size of page, 14.5x20.6 cm. 32 

lines to the page. 

Baker, Humphrey, " The Well spring of sciences which teacheth the per- 
fect worke and practife of "Arithmeticke, both in whole Numbers and 
Fractions : set forthe by Humphrey Baker, Londoner, 1562. And now once 
agayne perused augmented and amended in all the three parts, by the 
sayde Aucthour : whereunto he hath also added certain tables of the agree- 
ment of measures and waightes of divers places in Europe, the one with 
the other, as by the table following it may appeare." 
London, 1580; 1st ed., 1562. 

" In spite of the date 1562 on the title page, I find no edition before 1568. 
Indeed in the 1580 edition Baker says : ' Having fometime now twelve, 
yeres fithence (gentle reader) publifhed in print one Englifhe boke of 
Arithmetick ... I have been . . . requefted ... to adde fomething more- 
thereunto.' " D. E. Smith. 

Belli, Silvio, " Qvattro Libri Geometrici Di Silvio Belli Vicen 
tino. II Primo del Mifurare con la vifta. Nel Qvale STnsegna, Senza 
Travagliar con numeri, a mifurar facilif limamente le diftantie, Taltezza, e 
le profondita con il Quadrato Geometrico, e 'con altri ftromenti, de' quali 
facilmente fi puo prouedere con le Figure. Si mostra ancora vna belliffima 
via di retrouare la profondita di qual fi voglia mare, & vn modo induf- 
triofo di .mifurar il circuito di tutta la Terra. Gli Altri Tre Sono Delia 
Proportione & Proportionalita communi paffioni de Quanta. 

" Vtili, & necef farij alia vera, & ifacile intelligentia dell' Arithmetica della 
Geometria, & di tutte le fcientie & arti." 

Venice, 1595. 

9 



10 BIBLIOGRAPHY 

Belli, Silvio, " Silvio Belli Vicentino Delia Proportione, et Proportion- 
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Venice, 1573. 

Boetius, Anicius Manlius Torquatus Seuerkius, "Opera." Venice, 1491. 

Boetius, " Boetii Arithmetdca." Augustae, E. Rot dolt, 1488. 

Borgi, Piero, " Qui comeza la nobel opera de arithmeticha ne laquel se 
tracta tote cosse ameroantia pertinente facta 1 conipilata per Piero borgi 
da Venesia." Vcaksi 14^8; 1st ed., 1484. 

Brucaeus, Henricus, " Henrici Brucaei Belgae Mathematicarum exer- 

citationum Libri Duo." _ 

Rostock, 1575. 

Buteo, Joan., " Logistica. Quae & Arithnietica vulgo dicitur in libros 
quinque digesta : quorum index summatim habetur in tergo." 

Leyden, 1559. 

Calanderi, Philipi, " Philippi Calandri ad nobilem et studiosum Julianum 
Laurentii Medicem de Arithrnethrica opusculum." 

Florence, 1491. 

Cardanus, Hieronymus, " Hieronimi C. Cardani Medici Mediolanensis, 
Practica Arithmetice, & Mensurandi singularis. In quaque preter alias 
cotinentur, versa pagina demonstrabit." 

Milan, 1539. 

Cataneo, Girolamo, " Dell' Arte Del Misvrare Libri Dve, Nel Primo De' 
Qvale S'Insegna a mifurare, & partir i Campi. 

" Nel Secondo A' Misvrar le Muraglie imbottar Grani, Vini, Fieni, & 
Strami ; col linellar dell' Acque, & altre cose necessarie a gli Agrimenfori." 
Brescia (no date). 

Cataneo, Pietro, " Le Pratiche Delle Dve Prime Matematiche Di Pietro 
Cataneo Senese." 

,. Venice, 1567; 1st ed., Venice, 1546. 

! 

Champenois, Jacques Ghavvet, " Les Institvtions De L'Arithmetique De 
Jacques Chavvet Champenois, Professeur es Mafchematiques en quatre par- 
ties : auec vn petit Traicte des fractions Astronomiques." 

Paris, 1578. There is no evidence of an edition earlier than 1578. 

Chiarini, Giorgio, " Qvesta e ellibro che tracta de Mercatantie et vsanze 
de paesi." Florence, 1481. 

Colophon: " Finito ellibro di tvcti ichostvmi: caanbi: monete: . . . Per 



BIBLIOGRAPHY U 

me Francifco di Dina di Iacopo Kartolaio Fioretino adi x di Decembre 

MCCCCXXXI." 

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Ciacohi, Givseppe, " Regole Genarali D'Abbaco Gon Le Sve Dichiara- 
zioni, E Prove Seeondo L'Vso praticato da' piu periti Arimmetici ; Con. 
Vn Breve Trattato Di Geometris, E Modi del mifurare le superficie de' 
terreni, e corpi solidi. Deseritte Da P. Givseppe Ciaechi Fiorentino." 

Florence, 1675. 

Cirvelo, Petro Sanchez, " Tractatus Arithmeticae practice qui dicitur al- 
gorifmus. Venundantur Parrhifijs a johane Laberto eiufdem ciuitatis bib- 
liopola in stemate diui claudij manente iuxta gymnarium coquereti." 

Paris, 1513; 1st ed., 1495 (Treutlein). 

Clavius, Christopher, " Christophori Clavii Bambergensis E Societate 
IESV Epitome Arithmeticae practicae." 

1621 ; 1st ed., Rome, 1583. 

Clichtoveus, "Introductio Jacobi fabri Stapulefis in Arithmeticam Diui 
seuerini Boetii pariter Jordani. 

"Ars fupputaditam per calculos q? p notas arithmeticas fuis quidem reg- 
ulis elegater expreffa Iudoci Clichtouei Neoportuenfis. 

" Queftio haud indigna de numeroru , Aurelio Augustine 

" Epitome rerum geometricaruw ex Geometrico introductorio Carolio 
Bouilli, De quadratura circuli Demonstratio ex Campano. ,, 
Paris, c. 1506; 1st ed., Paris, 1503. 

de Muris, John, "Arithmeticae Speculativae Libri duo Joannis de Muris 
ab innumeris erroribus quibus hactenus oorrupti, & vetustate per me peri- 
erant diligenter emendati. 

" Pulcherrimis quoque exemplis. Formisq? nouds declarati & in usum 
studiosae iuuentutis Moguntinae iam recens excusi." 

Moguntiae, 1538. 

di Pasi, Bartholomeo, " Tariffa de i Pesi e Misure corrispondenti dal 
Levante al Ponente e da una Terra a Luogo adl' altro, quasi per tutte le 
Parti del Mondo. Qvi comicia la utilissima opera chiamata taripha laq val 
■tracta de ogni sorte de pexi e misure conrispondenti per tuto il mondo 
fata e composta per lo excelente et eximo Miser Bartholomeo di Paxi da 
Venetia." 

Venice, 1557; 1st ed., 1503. 

Finaeus, Orontius, " De Aritihmetica Practica libri quatuor : Ab ipsa 
authore vigilanter recogniti multisque accessionibus recens lo.cupletati." 

Paris, 1555; the 1st ed. was probably printed under the title "Arithmetica 
Practica libris quatuor absoluta" in 1525. 



I2 BIBLIOGRAPHY 

Gemma Frisius, "Arithmeticae Practicae Methodus Facilis, per Gemmam 
Frisium Medicum ac Mathematicum, in quatuor partes divisa." 
Leipsic, 1558; Leipsic, 1575; 1st ed., Antwerp, 1540. 

Ghaligai, Francesco, " Pratica D'Arithmetica." 

Florence, 1552. 
"This is a reprint of his Suma De Arithmeti'ca. Fireze m.ccccc.xxi." 

DeMorgan, (n.) p. 102. 
The title of the 1521 edition is given in the Boncompagni Bulletino, 13 : 
249. 

Gio, Padre, " Elementi Arithmetici Nelle scvole Pie." 

Rome, 1689. 

Grammateus, H., " Ein neu kiinstlich Rechenbuchlein uff alle kauffmann- 
schafft nach gemeinen Regeln de tre, Welsohenpractic." 

Frankfort, 1535. 

Heer, Johann, " Compendium Arithmeticae das ist : Ein neues kurtzes 
vfi wolgegrundtes Schul Rechenbuchlein/ von allerley Hauss : vnnd Kauff- 
mansrechnung/ wie die taglich furfallen mogen/ so per Regulam Detri vnd 
Practicam lieblich zu Resolvirn vnd auff zulosen sein? 

" Meinen lieben Discipulis, bene=:ben alien anfahenden Rechenschulern/ 
auoh sonsten Manniglich zu nutz/ also auffs fleissigste (So wol fur Jung- 
fraulein als Knaben) auss rechtem grund gestellet vnd inn Druck gegeben/ 
Durch Johann Heern Seniorem, Rechenmeistern vnd verordneten Visita- 
torn der Teutschen Schreib vnd Rechenschulen inn Niirmberg." 

Nuremberg, 1617. 

Huswirt, Johannes, " Enchiridion Novus Algorismi." 

Cologne, 1 501. 

Jacob, Simon, " Rechenbuch auf den Linien und mit Ziffern/ sampt aller- 
ley vortheilen/ fragweise/ Jetzt von neuem und zum neuntenmal mit vielen 
grundtlichen anweisungen/ oder Demonstration/ sampt derselben Vnder- 
richtung gemehret." 

Frankfort-on-the-Main, 1599; ist ed., Frankfurt, 1560. 

Jean, Alexander, "Arithmetique Av Miroir Par lequelle on peut (en 
quatre vacations de demie heure chacune) pratiquer les plus belles regies 
d'icelle. Mise en lumiere, Par Alexandre lean, Arithmeticien." 

Paris, 1637. 

Jordanus Nemorarius, "Jordani Nemorarii Arithmetica cum Demon- 

strationibus Jacobi Fabri Stapulensis - - - episdem epitome in Libros Arith- 

meticos D. Seuerini Boetii, Rithminachia (edidit David Lauxius Brytannus 

Edinburgensis)." ^ c 

& y Pans, 1496. 

Kobel, Jacob, " Zwey rechenbuchlin, uff der Linien vnd Zipher/ Mit eym 



BIBLIOGRAPHY 13 

angehenckten Visirbuch/ so verstendlich fiir geben/ das iedem hieraus 
on eifi lerer wol zulernen. 

"1 Durch den Achtbaren und wol erfarnen H. Jacoben Kobel Stat- 
schreiber zu Oppenheym." 

Oppenheim, 1537; 1st ed., 1514. 
Masterson, Thomas, " His first book of Arithmeticke." 

London, 1592. 

Masterson, Thomas, " His second book of Arithmeticke." 

London, 1594. 

Masterson, Thomas, " His Thirde booke of Arithmeticke." 

London, 1595. 

Maurolycus, Franciscus, " D. Francisci Mavrolyci Abbatis Messanensis, 
Mathematici celeberrimi, Arithmeticorum Libri Duo, Nunc Primum In 
Lucem Editi, Cum rerum omnium notabilium. Indice copiosissimo." 

Venice, 1575. 

Noviomagus, Joan, " De Numeris Libri II. Quorum prior Logisticen, 
& veterum numerandi consuetudinem : posterior Theoremata numerorum 
complectitur, ad doctissimum virum Andream Eggerdem professorem 
Rostochiensem." 

Cologne, 1344; 1st ed., Cologne, 1539. 

Pugliesi, "Arithmetica di Onofrio Pvgliesi Sbernia Palermitano." 
Palermo, 1670; 1st ed., 1654. 

Paciuolo, Lucas, " Suma de Arithmetica, Geometria, Proportioni et Pro- 

portionalita ." 

Tusculano, 1523 ; 1st ed., Venice, 1494. 

Peurbach, Georg von, " Elementa Arithmetices." 
Wittenberg, 1536. 

Perez de Moya, Jvan, "Arithmetica practica." 

Barcelona, 1703; 1st ed., Madrid, 1562. 

Raets, Willem, "Arithmetica Oft Een niew Cijfferboeck/ van Willem 
Raets/ Maesterichter. VVaer in die Fondamenten feer grondelijck ver- 
claert en met veel schoone queftien gheilluftreert vvorden, tot mit ende 
oorbaer van alle Coopliede ende leefhebbers der feluer Consten. 

" Met noch een Tractaet van de VViffelroede, met Annotatien verciert, 
door Michiel Coignet." 

Antwerp, 1580 (probably 1st ed.). (Original privilege dated May 22, 
1576.) 

Ramus, Peter, " Petri Rami Professoris Regii, Arithmeticae Libri Duo." 
Paris, 1577; 1st ed., Paris, 1555. 



I4 BIBLIOGRAPHY 

Recorde, Robert, "The Ground of Artes : Teaching the woorke and 
practife of Arithmetike, both in whole numbres and Fractions, after a more 
easyer and exaoter sorte, than anye lyke hath hytherto beene set forth : 
with divers new additions. Made by M. Roberte Recorde, Doctor of 

London, 1558; 1st ed., c. 1540 (DeMorgan, p. 22). 

Riese, Adam, " Rechnung auff der Linien und Federn/ auff allerley 
Handtierung/ Gemacht durch Adam Risen (auffs newe durchlesen/ und 
zu recht bracht)." 

Leipsic, 1571 ; 1st ed., Erfurt, 1522. 

Rudolff, Christopher, " Kunstliche rechnung mit der Ziffer und mit den 
zal pfennige/ sampt der Wellischen Practica/ und allerley vorteil auf die 
Regel de Tri. Item vergleighug mancherley Land un Stet/ gewicht/ Eln- 
mas/ Muntz ec. Alles durch Christoffen Rudolff zu Wien verfertiget." 
Wien, 1534; 1 st ed., 1526. 

Sacrobosco, Johann von, "Algorismus." 

Venice, 1523; 1st ed., 1488. 

Schonerus, Ioannes (Editor), "Algorithmus Demonstratus." 
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Stevinus, Simon, " Les Oeuvres Mathematiques de Simon Stevin de 
Bruges, Ou font inferees les memoires mathematiques. Le tout, correge 
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Scheubel, Johann, " De numeris et ddversis rationibus." 

Argent., 1540. 

Suevus, Sigismund, "Arithmetioa Historica. Die Lobliche Rechenkunst. 
Durch alle Species vnd furnembste Regeln/ mit schonen gedenckwirdigen 
Historien vnd Exempeln/ Auch mit Hebraischer/ Grichischer/ vnd 
Romischer Muntze/ Gev/icht vnd Mass/ deren in Heiliger Schrifft vnd 
gutten Gesohichte = Buchern gedacht wird/ Der lieben Jugend zu gutte er- 
kleret Auch denen die nicht rechnen konnen/ wegen vieler schonen His- 
torien vnd derselbigen bedeutungen lustig vnd lieblich zu lesen. 

" Aus viel gutten Buchern vnd Schrifften mit fleis zusammen getragen. 
Durch Sigismundum Sueuum Freystadiensem, Diener der H. Gottlichen 
Worts der Kirchen Christi zu Breslaw/ Probst zum H. Geiste/ vnd Pfarr- 
herr zu S. Bernardin in der Newstadt." 

Breslau, 1593. 

Tagliente, Giovanni Antonio and Girolamo, " LiBro DABACO Che in 
Segna a fare ogni ragione mercadantile, & pertegare le terre co Tarte dj la 
Geometris, & altre nobilifsime ragione ftra ordinarie co la Tariffa come 



BIBLIOGRAPHY 



15 



respondent* li pefi & monede de molte terre del mondo .con la inclita citta 
di Venegia. El qual Libro fe chiama Thefavro vniuersale." 

Venice, 15 15. 

Tartaglia, Nicolo, " La Prima Parte del General Trattato di Numeri, et 
Misure, di Nicolo Tartaglia, Nellaquale in Diecisette Libri si Dichiara 
Tutti Gli Atti Operativi, Pratiche, et Regole Necessarie non Solamente in 
tutta l'arte negotiaria & mercantile, ma anchor in ogni altra arte, scientia, 
ouer disciplina, doue interuenghi il calculo." 

Venice, 1556. 

Tartaglia, Nicolo, "La Seconda Parte del General Trattato Di Numeri, 

et Misure Nella quale in Vndici Libri Si Notifica La piv Ellevata, et 

Specvlativa Parte Delia Pratica Arithmetica, la qual e tutte regole, & oper- 
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Kunstrechnung/ grundlich beschrieben/ inn Frag und Antwort gestellet 
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Nuremberg, 1587. 

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liures. Ensemble un petit discours des Changes. Avec L'Art de calculer 
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regies & articles, par L'Autheur." 

Lyons, 1578; 1st ed., Lyons, 1566. 

Unicorn, Giuseppe, " De L'Arkhrnetica vniuersale, del Sig. Ioseppo Uni- 
corno, mathematico excellentissimo. Trattata, & amplificata con somma 
eruditione, e connoui, & isquisiti modi di chiarezza." 

Venice, 1598. 

Van Ceulen, Ludolf, " De Arithmetische en Geometrische fondamenten, 
van Mr. Ludolf Van Ceulen, Met het ghebruyck van dien In veele ver- 
scheydene constighe questien, soo Geometrice door linien, als Arithmetice 
door irrationale ghetallen, oock door den regel Coss, ende de tafeln sinuum 
ghesolveert." Leydeii; 



z6 BIBLIOGRAPHY 

Van der Schuere, Jaques, "Arithmetica, Oft Reken=const/ Verchiert 
met veel schoone Exempelen/ seer mit voor alle Cooplieden/ Facteurs/ 
Cassiers/ Ontfanghers/ etc. Gehmaeckt/ Door Jaques van der Schvere 
Van Meenen. Nu ter tij-dt Francpysche School-meester tot Haerlem." 

Haarlem, 1600. 

In the voor-Reden of the 1625 edition written by his son, Denys van der 
Schuere, it is stated that the book was published by the father " eerft in't 
Iaer 1600. Ende dit is al <Le vierde mael dat het gedrukt is." 

Wencelaus, Martin, T'Fondament Van Arithmetica: mette Italiaensch 
Practijck/ midtsgaders d'aller nootwendichste stucken van den Reghel van 
Interest. 

" Beydes in Nederduyts ende in Franchois/ met redelicke ouereenstem- 
minghe ofte Concordantien. Alles Door Martinvm VVenceslaum." 

Middelburg, 1599. 

Signs preface in Dutch, Marthin Wentfel; in French, Marthin Wentfle. 

Widman, Johann, " Behend und hupsch Rechnung uff alien Kauffman- 
schafften. Johannes Widman von Eger." 

Pforzheim, 1508; 1st ed., Leipsic, 1489 (Unger, p. 40). 

Although there was a 1500 edition, the 1508 edition was set up anew. 



CONTENTS 



CHAPTER I 

ESSENTIAL FEATURES OF SIXTEENTH CENTURY ARITHMETIC 

PAGE 

I. Productive activity of writers of that period 23 

1. Number of works and editions 23 

2. Causes of this awakening 23 

a. Revival of scientific interest 2^ 

b. Commercial activity 23 

c. Invention of printing ... 2^ 

II. Transitions through which the subject passed 24 

1. From the use of Roman symbols and methods to the use of 

Hindu symbols and methods 24 

2. From arithmetic in Latin to arithmetic in the vernacular . 24 

3. From arithmetic in Mss. to arithmetic in printed books . 24 

4. From arithmetic for the learned to arithmetic for the people. 24 

5. From arithmetic theoretic to arithmetic practical 24 

6. From the use of counters to the use of figures 24-29 

III. Nature of the subject matter of sixteenth century arithmetic. 29-168 

1. Definitions 29-36 

a. Of numbers, unity, and zero 29-32 

b. Of classes of numbers 32-35 

c. Of processes 35~36 

2. Processes with integers 36-77 

a. Notation and numeration ... 37-41 

b. Addition 4i~49 

c. Subtraction 49-57 

d. Multiplication 57-69 

e. Division 69-76 

f. Doubling and Halving 7^~77 

3. Denominate numbers 77-85 

4. Fractions - 85-110 

a. Definitions 85-93 

b. Order of processes 93~95 

c. Reduction 95~98 

d. Addition 98-102 

e. Subtraction 102-104 

f. Multiplication 104-107 

g. Division 107-110 

17 



1 8 CONTENTS 

PAGE 

5. Progressions 110-117 

6. Ratio and Proportion 117-121 

7. Involution and Evolution 121-127 

8. Applied arithmetic . . 127-168 

a. Number of writers of practical arithmetic compared with 

the number of writers of theoretic arithmetic . . . 127-128 

b. Examples of exceptional writers — Champenois, Suevus. 128-131 

c. List of business rules 131-132 

Rule of Three (Two and Five) 132-135 

Welsch Practice 135-138 

Inverse Rule of Three 138-139 

Partnership (with and without time) 139-141 

Factor Reckoning 142 

Profit and Loss 142-143 

Interest, Simple and Compound 143-145 

Equation of Payments 145-146 

Exchange and Banking 146-148 

Chain Rule 148-150 

Barter . 150-151 

Alligation 151-152 

Regula Fusti 152-153 

Virgin's Rule .... 153 

Rule of False Assumption, or False Position . . . 153-156 

Voyage 156 

Mintage 156-157 

Salaries of Servants 157 

Rents 157 

Assize of Bread . . 157 

Overland Reckoning 157-158 

d. Puzzles . . . 159-163 

e. Mensuration 163-168 

IV. Summary 168-170 

1. This was an important period in perfecting the processes of 

arithmetic 168 

a. The four fundamental processes with integers perfected. 168 

b. The four fundamental processes with fractions perfected. 168 

c. Arithmetic, geometric, and harmonic series (finite) fully 

treated . . 168 

d. Involution and evolution practically complete 168 

e. Tables for shortening computation were not unknown. 168 

f. Decimal fractions and logarithms were the only subjects 

not matured ... 168 

2. Constructive period for applied arithmetic 168-169 

a. Subject matter of applied arithmetic 169 

(1) Extent , 169 

(2) Rich in types 169 

b. Methods of solution of applied problems 169 



CONTENTS jg 

PAGE 

(i) Unitary analysis 169 

(2) Rule of Three . 169 

3. A comparison of the status of arithmetic at the beginning of 

this period with that at the close 169-170 

V. The place of arithmetic in the schools of that period 170-184 

A. In the Latin Schools 170-178 

1. The function of the Latin Schools 170 

a. To teach the Latin language 170 

b. To contribute to general culture 170 

2. Courses of study in the Latin schools 170-172 

a. Language — Latin 170-171 

b. Music 172 

c. Arithmetic 172 

d. Astronomy 172 

e. Geometry 172 

3. Writers of Latin School Arithmetics 172-174 

a. Profession—vocation 174 

b. Scholarship 174 

4. Character of the contents of Latin School Arithmetics. 174-177 

a. Prominence of pure arithmetic 174-175 

(1) Definitions 174-175 

(2) Classifications 175 

(3) Plans of organization 175 

b. Applications— chiefly artificial or traditional problems 176 

c. Absence of commercial arithmetic 176 

5. Reasons for teaching arithmetic in the Latin Schools. 176-178 

a. It was part of the classical inheritance 178 

b. It was supposed to contribute to general culture or to 

mental efficiency 178 

B. In the Reckoning Schools 178-184 

1. The rise of the Reckoning Master 178-179 

a. General duties. . 179 

b. Relation to schools and education , . 179 

2. The function of the Reckoning Schools 180 

a. To teach business methods . 180 

b. To teach commercial arithmetic 180 

3. Courses of study in the Reckoning Schools ...... 180 

a. Practice in reading and writing the mother tongue. 180 

b. Business customs and forms 180 

c. Arithmetic - . 180 

4. Writers of Reckoning School Arithmetics 180-183 

a. Profession — vocation 183 

b. Scholarship 183 

5. Character of the contents of the Reckoning School 

Arithmetic 183-184 

a. Meagre treatment of pure arithmetic 183 

(1) Little emphasis on definitions 183 



20 CONTENTS 



(2) Little emphasis on classifications 183 

(3) Processes confined to comparatively small num- 

bers — those used in practical applications . 184 

b. Prominence of applied arithmetic 184 

(1) Applications followed closely upon the pre- 184 

sentation of processes 184 

(2) Concrete problems often proposed before the 

required process had been developed . . . 184 

(3) Commercial problems and problems in men- 

suration were the chief applications .... 184 
6. Reasons for teaching arithmetic in the Reckoning 

Schools 184 

a. Because of its use to artisans 184 

b. Because of its use in commercial pursuits . , . . 184 

CHAPTER II 

EDUCATIONAL SIGNIFICANCE OF SIXTEENTH CENTURY ARITHMETIC 

I. Introduction 185 

II. Subject Matter . 186-201 

A. Kinds f 186 

1. Reckoning with counters and figures 186 

2. Properties of numbers 186 

3. Denominate numbers 186 

4. Business problems 186 

5. Amenity and puzzle problems 186 

B. Bases of selection 186-199 

1. Needs of the trader 186-190 

a. Commercial development tends to vitalize arithmetic 187 

b. Modern conditions will not revive the Reckoning 

Book 188 

c. The needs of the trader lead to improved methods 

of calculation 189 

d. Business needs condition the selection of denomi- 

nate number tables 190 

2. Needs of the scholar 190-196 

a. The modern disciplinary ideal demands concrete 

subject matter 191-193 

b. The culture ideal encourages the selection of subject 

matter with a many-sided interest 194-196 

c. The propaedeutics of arithmetic demand the reten- 

tion of certain theoretic matter . ■ 196 

3. Tradition 196-199 

This tends to perpetuate obsolete material i97 _I 99 

C. Plans of arrangement ....... 199-201 

1. By kinds of numbers 199 

2. By kinds of processes 199 

3. Modern needs are met by a combination of (1) and (2) 200-201 



CONTENTS 21 

PAGE 

III. Method 201-222 

A. Meaning 201 

B. Suggestiveness of sixteenth century arithmetic 201 

C. General principles of treatment of subject matter 201-202 

1 . The synthetic method not adapted to elementary arith- 

metic 202 

2. The analytic method may improve books, Stmt cannot sup- 

plant the'teacher 203 

3. The psychological method produces the best books in 

all particulars ■ . 203-204 

D. Details of development 204-222 

1. Definitions 204-205 

2. Notation .... 205-206 

a. The use of improved notations may be hastened . . 205 

b. Roman notation to thousands should be retained . 205 

c. Notation for large numbers is necessary and belongs 

to grammar school arithmetic 206 

3. Processes with integers 207-211 

a. In general, there is no best method for performing 

processes 207-208 

b. Artificial means for making number work interest- 

ing should not be abandoned . 208 

c. Incorrect language not an inheritance from six- 

teenth century arithmetic 208 

d. Methods of testing work are derived from the early 

arithmetics 208-209 

e. Number combinations are not all equally important. 209-210 

f. Explanations of processes should not be neglected. 210 

g. Books should explain mathematical conventions . 210 
h. Unabridged processes should precede abridged ones. 211 

4. Processes with fractions . . . . 211-214 

a. Common fractions are still necessary 211 

b. There are three ideas necessary to the concept of 

fractions 211-212 

c. Formal multiplication should precede formal addi- 

tion and subtraction 212 

d. The use of the word "times " in multiplication . . 212-213 

e. The two customary methods of division of fractions 

are related.. .... 213 

f. Fractions should be correlated with denominate 

numbers 213-214 

5. Denominate numbers 214-215 

a. Operations with compound numbers should be lim- 

ited to two or three denominations 214 

b. There is no good reason for continuing the practice 

of reduction from one table of denominate num- 
bers to another 214-215 

c. Denominate numbers should be presented under 

each process with integers and with fractions . . 215 



22 CONTENTS 

PAGE 

6. Applications 216-222 

a. Applications may be proposed as incentives for 

learning the processes 216-218 

b. Applications should be appropriate to the different 

school years 218-219 

c. Mensuration work should be graded 219 

d. Factitious problems have no place in elementary 

arithmetic 219-220 

e. Unitary Analysis, Rule of Three, and the Equation 

are related processes of solution 220-221 

f. The simple equation will become the leading 

method of solution 221 

g. Arithmetic has an interpretative function 221-222 

IV. Mode 223-225 

A. The heuristic mode suggests that oral work should develop 

new ideas 223 

B . The individual mode is apt to result in dogmatic teaching. 224 

C. The recitation mode finds no precedent in sixteenth cen- 

tury arithmetic 224 

D. The lecture mode has no place in elementary arithmetic . 224-225 

E. The spirit of the laboratory mode may be helpful in teach- 
ing arithmetic 225 

V. Summary 226-228 



CHAPTER I 

The Essential Features of Sixteenth Century 
Arithmetic 

The arithmetic of the last quarter of the fifteenth century 
and that of the sixteenth century show that great productive 
activity possessed the arithmeticians of that period. 1 Ap- 
proximately three hundred works were printed on this sub- 
ject, some of which ran through many editions, 2 This awak- 
ening was part of the great Renaissance and was due to the 
same causes; those influencing arithmetic most directly were 
the revival of scientific interest, commercial activity, and the 
invention of printing. The first cause was the incentive which 
led scholars to develop the science of figure reckoning; the 
second made a knowledge of casting accounts and of reckon- 

1 De Morgan (Arithmetical Books, pp. v-vi) says that from 1500 to 1750 
probably three thousand works on arithmetic were printed in all languages. 
He mentions fifteen hundred of them, but only seventy of the number 
printed before 1600 had been seen by him. It is probable, according to 
De Morgan, and by reference to Peacock's article in the Encyclopedia 
Metropolitana, that Peacock was familiar with a still smaller number. 

According to Kuckuck (Die Rechenkunst im sech7ihnten Jahrhundert, 
p. 16) over two hundred works on arithmetic were published in this period. 
He quotes Michel Stifel (1544) as saying that a new one came out every day. 

Riecardi (Bibliotica Mathematica Italiana, Vol. II, pp. 20-22), the great 
authority on the bibliography of Italian mathematics, gives one hundred 
and twelve Italian works under the title : " Trattati e Compendi di Arim- 
metiea." Of the extant works the number in German is about equal to the 
number in Italian. There are one-fourth as many Dutch, one-fourth as 
many French, one-fifth as many English, and one-tenth as many Spanish. 
Assuming that Riccardi's list is complete, and that the books lost bear a 
constant ratio to the number extant in all languages, one may conclude that 
there were approximately three hundred arithmetics printed before 1600. 

2 Professor Smith has found that Gemma Frisius's "Arithmeticae Prac- 
ticae Methodus Facilis " saw fifty-six editions before 1600, although Treut- 
lein (Abhandlungen, 1 : 18) found only twenty-five. 

Adam Riese's books in various combinations saw at least twelve editions 
before 1600. Unger, pp. 49-51. 

Recorde's and Baker's works in England enjoyed a similar popularity. 



24 



SIXTEENTH CENTURY ARITHMETIC 



ing exchange indispensable; and the third made the dissemin- 
ation of this knowledge possible. Another circumstance 
which encouraged both the development and the use of arith- 
metic was the expression of the subject in the vernacular, for 
hitherto a knowledge of theoretical arithmetic had been poss- 
ible to scholars only. In the sixteenth century arithmetics ap- 
peared in nearly all European languages, especially in Italian, 
French, German, Dutch, and English. 

Thus, the first century of printed arithmetics marks sev- 
eral important transitions : The transition from the use of the 
Roman symbols and methods to the use of the Hindu symbols 
and methods ; * from arithmetic expressed in Latin to arith- 
metic expressed in the language of the reader ; 2 from arith- 
metic in manuscript to arithmetic in the printed book ; 3 from 
arithmetic for the learned to arithmetic for the people ; 4 from 
arithmetic theoretic to arithmetic practical ; 5 and from the 
use of counters to the use of figures. 6 Although the Hindu 
numerals had been generally known to European mathema- 
ticians after the twelfth century, 7 the devotion to classical 

1 E. g., compare Jacob Kobel, Zwey rechenbuchlin uff .der Linien und 
Zipher/ - - - (1537) with Robert Recorde, The Ground of Artes - - - 
(c 1540). 

2 E. g., compare Gemma Frisius, Arithmeticae Practicae Methodus Facilis 

(1558) with Ian Trenchant, L'Arithmetique, Departie en trois livres 

(1578). 

3 The Treviso Arithmetic (1478), so called from the place of printing, is 
the earliest printed arithmetic known to exist. 

4 E. g., compare Hieronimus Cardanus, Practica Arithmetice & Mensur- 
andi singularis - - - (1539) with Willem Raets, Ein niew Cijfferboeck 
- - - (1580). 

B E. g., compare Joannis <de Muris, Arithmeticae Specuilativae Libri duo 
(1538) with Adam Riese, Rechnung auff der Linien und Federn/ - - - 
(i57i). 

6 Adam Riese (1522) and Robert Recorde (1557) show this transition 
by treating both systems of reckoning in their books. 

7 M. F. Woepcke, Propogation des chiffres indiens, Journal Asiatique, 
6 ser, t. i, pp. 27, 234, 442. 

Ch. Henry, Sur les deux plus anciens traites Francois d'Algorisme et de' 
Geometrie, Bone. Bull., 15 149. 

Freidlein, on John of Seville and Leonardo of Pisa, Zeitschrift der 
Mathematik und Physik, Band 12, 1867. 

A. Kuckuck, Die Rechenkunst im sechzehnten Jahrhundert (1874). "I« 






THE ESSENTIAL FEATURES 25 

study and the neglect of science tended to perpetuate the Ro- 
man numerals. Not until the introduction of printing did a 
full comparison of the Roman and Hindu arithmetics find ex- 
pression, and a working knowledge of the latter sift down to 
the common people. 

Owing to the Romans' lack of appreciation of pure science 
and to the awkwardness of their system of notation, their 
contribution to arithmetic was small. Arithmetic, in the form 
of the ancient Logistic, was of use to them chiefly in making 
monetary 'calculation, for which they used some form of the 
abacus. The nations of Europe received as a legacy from 
the Romans the Roman numerals and the art of reckoning 
with counters. This art, also called line reckoning, was wide- 
spread in the fifteenth century, excepting in Italy. The cal- 
culations were effected by means of parallel lines drawn on a 
board or table, and by movable counters or disks placed upon 
them. The lines taken in order from the bottom upward rep- 
resented units, tens, hundreds, thousands and so on. The 
spaces taken in the same order represented fives, fifties, five 
hundreds, and so on, the space below units being used for 
halves. The table on the next page shows Riese's explanation 
of the lines and spaces : 1 

>einer Regensburger Chronik von 1167 feefinden sich die Zahlen von 1-68, 
-aber nur wie zur Uebung geschreiben. In Schlesien kommen sie erst im 
Jahre 1340 vor. In einem Notatenbuch 'des Dithmar von Meckelbach aus 
-der Zeit Kaiser Carls IV stehen die Ziffern 1-10." Pp. 4-5. 

See also the following general references : 

Treutlein, Geschichte unserer Zahlzeichen (1875 — Program Gymn. Karls- 
ruhe), Bd. 12, 1867. 

Gerhardt, Geschichte der Math, in Deutsehland (1877). 

Freidlein, Die Zahlzeichen und das elementare Rechnen der Griechen, 
Romer u. des christl. Abendlandes vom 7. bis 13. Jahrhundert, 1869. 

WMdermuth, Rechnen, in Schmids Encyklopadie, Bd. 6. 

1 Adam Riese, Rechnung auff der Linien und Fed'ern/ (1571 ed.), 

iol. Aiij recto. 



26 



SIXTEENTH CENTURY ARITHMETIC 



IOOOOO 

50000 

IOOOO 

5000 
IOOO 

500 
IOO 

50 

10 

5 
1 

ji 



Hundert tausent. 

Funfftzig tausent. 

Zehen tausent. 

Funf tausent. 

Tausent. 

Funf hundert. 

Hundert. 

Funfftzig. 

Zehen. 

Funff. 

Eins. 

Ein halbs. 



Cross lines were drawn dividing 
the board into sections, which 
could be used for different de- 
nominations of numbers, for the 
addends in a problem of addition, 
for the minuend and the subtra- 
hend in subtraction, or for any- 
other sets of numbers. 

The illustration 1 shows four 
compound numbers arranged for 
addition and the result expressed 
in floren, groschen, and denarii. 

Addition and subtraction with 
counters are evidently quite easy. 
In fact, line reckoning was often 
recommended as preferable to the 
processes with Hindu numerals. 2 



1R«wfo wmino Tttuo ptctmias mnrno t>ata$ credttot Hot 
*i'tll5jnc vtnqjcupiwt) audcrt (lifpiriotw? addition! con 
fultre hat Sun rmmcnaddcntorrtrefoltabit ©onjitia. 
ft Gf 9> 



#**M 



•&&&+■ 



-%%®- 



fr-pacjfecampinunmaroddnetuKpftfffrfmflutt Sub* 
fequai a fubtilwt campite . 

ft «f a 



• 
-*•* 



4|f2trtj(Ia#ra; additwne nootnaria pmo txptrietis ccUcctU 
one m q qm'cqd miii» neucnarto Cupd? ,pt>a amdlarar.qtf tfi 
to f ntelligert opoitct*$n nomsrie addcndts. 9.rp»rfim colle 
ctiroctoi8abtc[Jcurquocieti9r<pcrta.gi[o cjrptdito qxtic* 
qoidniimcr i Tub 9 vlnmo ranaTcnr id <id .pbam prwert'ne* 
ctflirdl auc ft addfdotum .pba.tocms rumniefimiUe mr a* 
beitunne 9 talicjgrnio ejclnina ?fpaci)8 Ofducrt. aru addi* 
tuftt Sin au t .ignanfe tfnms ef d vt fu mme tffttt crtdatnr 
vnCgfcmiinTjcambupjcbtadtcratitur* 



All that is necessary, after 



1 From Balthazar Lidht's Algorismus Linealis (1501 ed.). 

2 Rudolff, Kunstliche rechnung mit der Ziffer und mit den zal pfennige/ 
- - - (1526), "Das die vier spezies/ auff den linien durch viel ringere 
ybung auff der Ziffer gelernt werde/ mag ein yeder aus obenanzeigter vnter- 
weisung bey jm selbst ermessen. - - - Warlich was Fiirsten vnd Hernti 
Rentkamer/ vrbarbucher/ register/ aussgab/ empfang/ vnd ander gemeine 
hausrechnung belangt/ dahin ist sue am bequemisten/ zu subtilen rech- 
nungen zum dickermal seumlich." 

See also : 

Sterner, Ges>chichte der Rechenkunst, I Teil, pp. 218, 219. 

Kastner, Geschichte der Mathematik, Bd. I, p. 42. 

Knott, C. G., The Abacus in its Historic and Scientific Aspects, Trans- 



THE ESSENTIAL FEATURES 2 J 

having expressed the numbers, is to shift or set them so as to 
express the sum or difference of the numbers found on each 
line. The processes of multiplication and division are more 
complex; so much so, that, when the multiplier or divisor 
contains more than one figure, the Hindu algorisms are far 
superior. Several modern authorities give detailed explana- 
tions of these processes. 1 

The transition from line reckoning to reckoning with the 
Hindu numerals, or pen reckoning, as it was often called, was 
slow. So difficult was it to abandon the old line and space 
idea and the Roman symbols, that writers mixed the old and 
the new symbols in calculation. This is plainly shown in 
Toilet reckoning 2 (tabular calculation), a process used in a 
few early works, as in the Bamberg Arithmetic and in the 
arithmetics of Widman and Apianus. For example, the prob- 
lem : " What is the cost of 4,367 tb. 29 lot 3 quintl of ginger 
at 16 shillings a pound?" is solved thus by Widman : 3 

actions of the Asiatic Society of Japan, vol. xiv, part i, pp. 19, 34 (Yoko- 
hama, 1886). 

1 Knott, C. G., The Abacus in its Historic and Scientific Aspects, pp. 18, 
45-67. See preceding note. 

Ku'ckuck, A., Die Rechenkunst im sechzehnten Jahrhundert, pp. 10-13 
(Berlin, 1874). 

Villicus, Geschichte der Rechenkunst, pp. 68-76 (Wien, 1897). 

Leslie, Philosophy of Arithmetic, under "Palpable Arithmetic" (Edin- 
burgh, 1820). 

A more accessible work to many is : 

Brooks, Philosophy of Arithmetic, pp. 115, 160 (Lancaster, Pa., 1901). 

For the historical development of the abacus see : 

Cantor, M., Vorlesungen iiber Geschichte der Mathematik, Vol. I (Leip- 
sic, 2d ed., 1894). 

Also articles by Boncompagni in Atti dell' Accademia pontificia de 
nuovi Lincei. 

2 P. Treutlein, Abhand'lungen Zur Geschichte Der Mathematik, vol. 1, 
p. 98 (Leipsic, 1877). 

3 Johann Widman, Behend und hupsch Recbnung (1508 ed.). " Es hat 
einer kaufft 4367 lb' Ingwer 29 lot 3 quiintl/ ye 1 lb' fur 16 p in gold- 
setz also." Fol. Ei verso, Eii recto. 

(A mistake was evidently made in this edition, since the problem reads 
"13 shillings a pound " in the early editions.) 



28 



SIXTEENTH CENTURY ARITHMETIC 



4M 
3C 
6X 
7 lb' 

2X 

9 lot 

3 quintfi 



13000 p 

1300 p 

12,0 p 

130/32 p 

13/32 /> 

13/128 /> 



4M 
3C 
6X 
7 lb' 

2 X 
9 lot 

3 quitl 



52000 

3900 

780 

91/ 

260/32 

117/32 

39/128 



facit 



2600 

195 

39 floren 
4 flap 
8 4/32 

3 21/52 
39/128 



The first column at the left is the multiplicand, 4 thousand, 3 
hundred, 6 tens and 7 lb. ; 20 lot 'and 9 lot and 3 quintl. The 
lot and the quintl are first expressed as fractional parts of a 
pound. The Roman symbols, M, C, X, at the right of this 
column are superfluous, since the figures, 4, 3, 6, placed above 
one another designate by their positions the orders for which 
they stand. The columns beginning 13000 p form the multi- 
plier, 13, set down for each order. In modern work the 4000 
would be multiplied by 13, but here 4 is multiplied by 13,000. 
The fourth group represents the results of the multiplications. 
The numbers in the last column at the right express these re- 
sults in florins and shillings. More than half a century after 
Widman, Robert Recorde, in the later editions of his " Ground 
of Artes," says that before studying arithmetic proper the 
Roman numerals must be learned. 1 

A method of calculating, or a mnemonic to assist in abacus 
work, called Finger Reckoning, was explained by a few writ- 
ers of this period. 2 But as the method was then obsolete in 

1 Robert Recorde, T/he Ground of Artes (1594 ed.). " Before the intro- 
dlucbion of Ariithmeticke, it were very good to have fome vnderftandiing 
and knowledge of thefe figures and notes:" 

This- is followed by a table of Roman numerals with the corresponding 
Hindu numerals and the corresponding words, as : 

one 
two 
three Fol. Bviii verso. 

2 Noviomagus, De Numeris Liibri II, Cap. XIII. 

Paciuolo, Suma de Arithmetica Geometria Proportion! et Proportionalita 
(1494 ed.), fol. 36 verso, or Eiiij verso. (He does not explain, but gives 
a page of pictures.) 

Andres, Sumario breve de la practica de la arithmetica, Valencia (1515). 

Tagliente, Libro de Abaco, Venetia, M. D. XV (1541 ed.), fol. Aiii verso. 

Apianus, Ein newe vnd wolgegrundte vnderweysung aller Kauffmans- 
rechnung, Ingolstadt (1527). 

Aventinus, Abacvs at qve vetvstissima, vetervm latinorum per digitos 



i 


1 


ii 


2 


iiii 


3 



THE ESSENTIAL FEATURES 29 

Western Europe, it has no significance here. Accounts of 
this method are given in standard authorities. 1 

In order to exhibit the essentials of the arithmetic of this 
period briefly and systematically, it is best to treat it by topics, 
as was the common practice of its authors. 

DEFINITIONS 

It was the common practice among Latin School writers, 
especially among those who> were influenced by the works of 
the Greeks on theoretic arithmetic, to begin with a formidable 
list of definitions. 

Definitions of Number, Unity, and Zero 
Number was generally defined thus : " Number is a collection 
of units." The following is Paciuolo's definition : 2 "Num- 
ber is a multitude composed of units. Aristotle says, if any- 
thing is infinite, number is, and Euclid in the third postulate 
of the seventh book says that its series can proceed to infinity, 
and that it can be made greater than any given number by add- 
ing one." 

The meaning of unity caused writers much concern and was 
variously defined, as appears from the following : 

1. Unity is the beginning of all number and measure, for as 
we measure things by number, we measure number by unity. 3 

manusqj numerandi (quin etiam loquendi) cofuetudo, Ex beda cu picturis 

et imaginibus Ratispone (1532). 

Moya, Traitado 'de Matematicas, Alaaba, 1573 (1703 ed., chap. ix). "Trata 
dela orden que los antiquos timiero en eStar con los dedos de las manos, 
y otras partes del cuerpo." 

1 Leslie, Philosophy of Arithmetic (Edinburgh, 1820), p. 101. 
Villicus, Geadhichte der Rechenkunst (1897), pp. 10-14. 

Stoy, H., Zur Geschichte des Rechenunterrichts, I. Teil, § 9, p. 47. See 
plates at end of volume. 

Sterner, Geschichite der Rechenkunst, I. Teil, p. J7. 

Cantor, Vorlesungen uber Geschichte der Mathematik (1900 ed.), Bd. I, 
see index. 

2 Paciuolo, Suma de Arithmetica Geometria Propoirtioni et Proportion- 
alita (1523 ed.). " Numero : e (fecondo ciafchuno philofophante) vna 

moltitudine de vnita copofta : Ariftotile dike : eioe. Si quid infinitum 

eft : numerus eft. E per la terza petitione del f eptimo de Euclide : la f ua 
ferie in infinito potere prooedere: 2 quocuq? numero dato : dari poteft 
maior: vnitatem addendo." Fol. A i recto. 

3 Joan Nov/iomagus, De Numeris Libri II (1544). 



30 SIXTEENTH CENTURY ARITHMETIC 

2. Unity is not a number, but the source of number. 1 

3. Unity is the basis of all number, constituting the first 
in itself. 2 

4. Unity is the origin of everything. 3 

This difference of opinion as to the nature of unity was not 
new in the sixteenth century. The definition had puzzled the 
wise men of antiquity. 4 Many Greek, Arabian, and Hindu 
writers had excluded unity from the list of numbers. But, 
perhaps, the chief reason for the general rejection of unity 
as a number by the arithmeticians of the Renaissance was 
the misinterpretation of Boethius's arithmetic. Nicomachus 
(c. 100 A. D.) in his Apidfiquitw ptp?ua dv? had said that unity was 
not a polygonal number and Boethius's translation was sup- 
posed to say that unity was not a number. 5 Even as late as 
1634 Stevinus found it necessary to 1 correct this popular error 
and explained it thus : 3 — 1=2, hence 1 is a number. 6 

1 Gemma Frisius, Aritbmeticae Practicae MJethodus Faoilds (1575 ed.). 

" Numerum autem vocant multitudinem ex unitatibus conflatam. Itaque 

unitas" ipsa numerus nan erit, sed numerorum omnium principium." Fol. 

A_ verso. 
5 

Jacques Chavvet Champenois, Institvtions De L'Arithmetique (1578 ed.). 
" Vn, qui n'est pas nombre, mais comencement de nombre, & origine de 
toutes chofes, ." Page 3. 

Humphrey Baker, The Well Spring of Sciences (1580 ed.). "And 
tiherfore an vnitie is no number, but the begining and originall of number, 
as if you doe multiplie or deuide a vnite by it felfe, it is refolued into 
itfelfe without any increafe. But it is in number otherwife, for there 
can be no number, how great foeuer it bee, but that it may continually bee 
encreafed by adding euermore one vnitie vnto the fame." Fol. Bi recto. 

2 Franciscus Mavrolycus, Arithmeticorum Libri Duo (1575). "Unitas 
est principium & oonstitutrix omnium numeromm, constituens autem im- 
primis seipsam. ,, Page 2. 

3 See definition by Champenois in note 1 above. 

4 In Plato's Republic we find : " To which class do unity and number be- 
long?" Monroe's Source Book, p. 203. 

5 Weissenborn, H., Gerbert, Beitrage zur Kennitniss der Mathematiik des 
Mittelalters, p. 219 (Berlin, 1888). 

6 After reviewing the various arguments which history has handed down, 
Stevinus says : " Que l'unitie est nombre. II est notoire que Ton diet vul- 
gairement que 1' unite ne soit point nombre, ains seulement son principe, 011 



THE ESSENTIAL FEATURES 



31 



Zero was referred to merely as a symbol used in connection 
with the digits to express number. When taken alone it was 
said to have no meaning. The prevalence of nulla, nulle and 
rein in the terms used to express it is suggestive of this 
meaning. 1 

So far were the mathematicians of that period from the 
conception of number as a continuum that they emphasized 
the difference between continuous and discrete quantity and 
limited arithmetic to the latter domain. 2 So long as they 

commencement & tel en nombre come le point en La ligne; ce que nous 
nions, & en pouvons argumenter enter en cefte forte : 
La partie eft de mefme matiere quest fon entiefr, 
Vnite eft partie de multitude d'unitez; 

Ergo Vuniiie eft de mefme matiere qu'eft la multitude d'unitez; 
Mais la matiere des multitude d'unitez est nombre, 
Doncques la matiere d'unite eft nombre. 
Et qui le nie, faict comme celuy, qui nie qu' une piece de pain foit du pain. 
Nous pourrions auff i dire ainfi : 

Si du nombre donne Von ne foubftraict nul nombre, le nombre donne* 

demeure, 
Sent trois le nombre donne, & du mefme foubftrayons un, qui n'est 

point nombre comme tu veux. 
Doncques le nombre donne demeure, e'eft a dire qu'il y reftera en- 
core trois, ce qui eft abfurd. Fol. A recto. 

1 The various names for the symbol o in this period were : Zefiro and 
nulla, Piero Borgi (1488 ed.) ; cero and nulla by Paciuolo (1523 ed.) ; 
nulla, Rudolff (1534 ed.) ; cyphar, Recorde (1540) ; circolo, cifra, zerro, 
nulla, Tartaglia (1556) ; cyphram, Gemma Frisius (1558 ed.) ; circulus, 
Ramus (1577 ed.) ; nulle or zero, Trenchant (1578 ed.) ; nul and rein, 
Champenois (1578) ; ciphar, Baker (1580 ed.) ; and nullo, Raets (1580). 
Buteo (1559) says that the zero could be called omicron because of its form. 

The oldest manuscript actually known to have the zero bears the date 
738 A. D., by Jaika Rashtrakuta. 

2 Unicorn, De L'Arithmetioa vniuersale (1598 ed.). "Quantity is divided 
into two classes : continua and discreta. La quantita continua is for- 
mally defined to be that of which the terminus of every part joins ,bhe 
terminus of another part, a c b As, for example, in the line 

ob, the point c is the terminus of the part ac, and this is 
also a terminus of the part be, and a common terminus. 
Or in the surface abed, the line ef is the common ter- 
minus which divides it into two parts. Of this division 
of quantity there are five kinds : lines, surfaces, solids, 
place and time. The treatment of these belong to geometry. 

Quantita discreta is defined to be such that no part is joined to another 



32 SIXTEENTH CENTURY ARITHMETIC 

held to this limitation, they could never think of the one-to- 
one correspondence of numbers to points on a line. 1 The 
Greek method of representing surds by lines was well known, 
and it would have been easy to arrive at the conception of 
filling in the points of a line with numbers, had not continuity 
been excluded. 

Definitions of Classifications 

The definition of number was followed by definitions of the 
various classifications of numbers. The following taken from 
Paciuolo will illustrate : 2 

A number is prime when it is not divisible by any other in- 
tegral number but one and the number itself. Otherwise it: 
is composite. Examples of primes: 3, 7, 11, 13, 17, etc. 
Examples of composites : 4, 8, which is 2 X 4, 12, 14, 18, etc. 

Lateral or linear numbers. Different numbers which may 
be multiplied together, as 3 and 4, 6 and 8, compared to the 
sides of a rectangle. 

common part of another quantity, as a number. For example, in numbers 
with periods containing three orders (the usual method of numeration), 
the last number is the last of that group, and is not the beginning of the 
next group. 

Another difference between quantita discreta and quantita continua is 
that the continua is divisible ad infinitum and the discreta is increasable 
ad infinitum. 

The quantita continua is divided into mobile and immobile, and by im- 
mobile is meant the earth, and by mobile the heavens. Under the immobile 
is included geometry, and under mobile, astrology. 

Two other kinds of quantity : that which has position, as the solid, con- 
tinuous thing ; the other which has not position, as time, which is constantly 
passing, and water and other liquids, which have not position but which 
are limited by other things, as by the vessel containing the water. 

_ . f mobile — cielo. TT _ . .„, 

I. Contmua { immob j le -t e rra. 3- Hauete posit.oe. 

Quantita U > 

1 -r-x. ( numero. -_„ , 

(^ 2. Discreta -j .. 4. No hauete positione. 

Fol. A 2 recto. 

1 Dedekind, R., Essays on the Theory of Numbers (Beman's translation^ 
Chicago, 1901). 

2 Paciuolo, iSumma (1523 ed.), fol. Ai recto, Aij verso et seq. 



THE ESSENTIAL FEATURES 



33 



Superficial {plane) number. The product of two linear 
numbers, as 12 from 3 X 4, 48 from 6X8. 

Square number. The product of two similar numbers, as 
9 from 3 X 3> l6 from 4 X 4> 25 from 5 X 5> etc - 

^o/id number. The product of three linear numbers, as 12 
from 2X3X2. 

Cwfo'c number. The product of three equal numbers, as 8 
from 2 X 2 X 2, 27 from 3 X 3 X 3» 6 4 from 4X4 
X 4, etc. 

Triangular numbers. Those which commence with unity t 
and which increase upward in the form of a triangle by add- 
ing a unit, always keeping the sides equal. 1 

Besides this there are pentagonal numbers, etc. 

Circular numbers, as 5 and 6 ; so called because each multi- 
plied by itself to infinity always gives a product ending in 
itself, as 5 X 5 = 25 X 5 = 125 X 5 = 625 - - -; 6 X 6 
= 36 X 6 = 216 X 6 = 1296 . 

Defective numbers are those the sum of whose factors is less 
than the number itself, as 8 and 10. 8 = 4X2X154 + 
2 + 1 = 7; 10 =5X2X1; 5+2 + 1=8. 

Superfluous numbers. Those the sum of whose factors is 
more than the number itself, as 12, 24, etc. The factors of 
12 are 6, 4, 3, 2, 1 ; 6 + 4 + 3 + 2 + 1 = 16: factors of 
24 are 12, 8, 6, 4, 3, 2, 1 ; the sum of these is 36. 

Perfect numbers are those the sum of whose factors equals 
the number itself; e. g., the factors of 6 are 3, 2, 1, and their 
sum 1 is 6; the factors of 28 are 14, 7, 4, 2, 1, and their sum 
is 28. 

The manner of designating various ratios was also peculiar 
and elaborate. For example, the relation of any two num- 
bers whose ratio is 1^ to 1 was called sesquialteral, meaning 
that the antecedent contains the consequent once and one-half 



1 See marginal illustrations in Paciuolo. 

2 The sesquialtera stop of an organ which furnishes the perfect fifth in- 
terval, 1 : 1^2, is named from 'this old Greek ratio. 



34 SIXTEENTH CENTURY ARITHMETIC 

Similarly the ratio 

\Yz : i, or 4 : 3 was called sesquitertial. 

ij4 : 1, or 5 : 4 was called sesquiquartal, and so on. 
Of all the pairs of numbers that can result in sesqui ratios, 
the antecedents were called super particular is and the conse- 
quents subsuper particulars. x 

When the integral part of the ratio is greater than 1 the 
above ratios were preceded by the corresponding adjectives, 
thus the ratio 

2}/2 : 1 was called duplex sesquialteral. 

2^3 : 1 was called duplex sesquitertial. 

3>4 : 1 was called triplex sesquiquartal, and so on. 

When the fraction in the ratio exceeds a unit fraction, the 
ratio was named according to the numerator, thus the ratio 

i 2 A : 1 was called superbipartiens. 

iYat '. 1 was called supertripartiens, and so on, meaning ex- 
cess by two parts, by three parts, and so on. 

The prefix sub was used to designate the inverse of the above 
ratios, thus the ratio 

1 : 1% was called subsuperbipartiens. 

1 : \Ya was called subsupertripartiens. 2 

The reason for emphasizing these peculiar classifications of 
number is not manifest. Legendre 3 explained the prominence 
given to the subject on the ground that its study becomes a 
sort of passion with those who take it up. This phase of 

1 A discussion of the classifications of Nieomachus may be found in 
Gow, History of Greek Mathematics (Cambridge, 1884), page 90. 

2 After so much of explanation, a characteristic remark of DeMorgan 
will be appreciated : " For some specimens of the laborious manner by 
which the Pythagorean Greeks, in the first instance, and afterwards Boe- 
thius in Latin, had endeavored to systematize the expression of numerical 
ratios, I may refer the reader to the article Numbers, old appelations of, 
in the Supplement to the Penny Cyclopedia (London, 1833-43). If I were 
to give any account of the whole system, on a scale commensurate with the 
magnitude of the works written on it, the reader's patience would not be 
subquatuor decupla subsuperbipartiens septimas, or, as we should now say, 
seven per cent of what he would find wanted for the occasion." DeMor- 
gan, Arithmetical Books, p. xx. 

3 Legendre, Theories des Nombres (1798), preface. 






THE ESSENTIAL FEATURES 35 

arithmetic had its origin in the products of the Pythagorean 
School, was expounded by Nicomachus, 1 and was communi- 
cated to the scholars of the Renaissance by Boethius's trans- 
lation 2 of Nicomachus. Among the works of the sixteenth 
century which treated this subject, that of Maurolycus 3 is 
noteworthy both for its exposition of the Greek classifications * 
and for its doctrine of incommensurables. 

Writers on commercial arithmetic in the sixteenth century 
introduced their works by definitions of arithmetic, quantity, 
and number, but discarded the fanciful theory of numbers so 
attractive to theoretic writers. 

Definitions of Processes 
Each process was defined when first introduced, which was 
in connection with integers, since most of the writers treated 
the four fundamental processes with integers before doing so 
with fractions and denominate numbers. These definitions 
related to integers only, and often failed to have meaning when 
applied to fractions. Addition was generally defined as the 
collection of several numbers into one sum, 5 and subtraction 
as taking a smaller number from a larger one. Q Certain writ- 
ers 7 improved on this, and even recognized subtraction to be 
the inverse of addition. 8 Multiplication was generally defined 

1 See page 34, note 1. 2 See page 34, note 2. 

3 Mavrolycus, Franciscus, Arathmeticorum Libri Duo (1575), p. 3. 
*Jordanus (1496 ed.) also is noted for its extensive treatment of the 
Greek properties of numbers. 

5 E. g., Trenchant, L'Arithmetique (1578), "Aiouter, eft affembler plu- 
fieur nomibres en une fomme," «fol. B 4 recto; and Clichtoveus, Ars fuppu- 
tadi - - - (Paris, c. 1507). "Additio eft multorum numeroa figillatim 
fumptorum in unam fummam collectio." Fol. b iiij recto. 

6 E. g., Tartaglia, General Trattato di Numeri (1556). " Sottare non e 
altro, che duoi proposti numeri, inequali saper trouare la loro differentia, 
cioe quanto che il maggiore eccede il menore." Fol. Bvi verso. 

7 E. g., Tonstall, De Arte Supputandi (1522). " Subducto numerorvm 
est minoris numeri a maiore; uel aequalis ab equale fubtractio." Fol. E 
recto. 

8 Clichtoveus, Ars fupputadi (c. 1507). " Subftractio est numeri minoris 
a majori subduotio. Et additioni ex opposito respondet." Fol. biiii verso. 



36 SIXTEENTH CENTURY ARITHMETIC 

as repeating one number as an addend as many times as there 
are units in another, 1 a definition not directly applicable to 
fractions without modification. Division was generally defined 
as -finding how many times one number is contained in another/ 
the partitive phrase being often included. Its relation to sub- 
traction was also recognized. 3 

PROCESSES WITH INTEGERS 

The writers of that period did mathematics a service in re- 
ducing the number of processes in arithmetic. The processes 
were commonly called Species, 4 due to the influence of the 
Latin manuscripts. In 1370 Magistro Jacoba de Florentia 
gave 9 Species, the number common in mediaeval times, viz. : 
numeratio, additio, subtratio, duplatio, mediatio, multiplication 
divisio, progressio, et radicum extractio. The Hindus, ac- 
cording to the Lilavati, 5 had eight processes, which were in- 
creased to ten by the Arabs, who added Mediatio and Duplatio. 
These latter are found in El Hassar (c. 1200), and probably,, 
according to Suter, are of Egyptian origin. Their presence 
in mediaeval Latin manuscripts is due to the influence of Al 
Khowarazmi. 6 In the sixteenth century the number ranged 

1 Gemma Frisius, Arithmetlcae Practicae Methodus Faoills (1575 ed.). 
" Multiplicare est ex ductu vnius numeri in alteram numerum producere, 
qui toties habeat in se vnum multiplicantium, quoties alter vnitatem, Hoc 
est, Multiplicare est numerum quemcumq? aliquoties aut mul-toties, exag- 
gerare." Fol. B 2 verso. 

2 Trenchant, L'Arithmetique (1578). " Parttir, eft chercher quantes foys 
vn nombre, contient 1'autre." Fol. D 2 recto. 

3 Ramus, Aritbmeticae Libri Duo (1577 ed.). "Divisio est, qua divisor 
subductitur a dividendo quoties in eo continetur & habetur quotus." Fol. 
A vii verso. 

An interesting comparision of definitions current in the seventeenth cen- 
tury is found in DeMorgan, Arithmetical Books, pp. 59-61. 

4 The origin of the word " species " has been traced to the Greek word 
It-doq, meaning member of an equation. This word appeared in rules for 
adding to and subtracting from the members of an equation, and was 
translated into Latin as species, which later came to be used to designate 
all of the fundamental processes of arithmetic. Cantor, Vorlesungen uber 
Gesehichte der Mathematik (3d ed., 1900), Bd. I, p. 442. 

5 The work of Bhaskara, a Hindu writer (c. 1200 A. D.). 

6 An Arabian mathematician (c. 800 A. D.). 



THE ESSENTIAL FEATURES 



37 



from nine to five. Some writers excluded extraction of roots, 
others progressions also. Piero di Borgi argued that the 
number should be reduced to seven in order that it might cor- 
respond to the number of gifts of the Holy Spirit. 1 Later in 
(the century duplatio (multiplying by 2) and mediation (divid- 
ing by 2) were excluded, reducing the processes to numera- 
tion and the four fundamental operations recognized to-day. 2 
It is probable that numeration was not always included, in 
which case the number would be four. " The lack of agree- 
ment with reference to the number of Species finds its ex- 
planation in the circumstance that they fail to define the idea 
of Species. Gemma Frisius is the only one who attempted a 
definition : ' Moreover, we call certain kinds of operations with 
numbers Species.' " 3 But this is too indefinite to give any 
basis of selection. 4 

Notation and Numeration 
The object of numeration was to teach the reading of num- 
bers written in the Hindu notation. For a hundred years after 
the first printed arithmetic many writers began their works 
with the line-reckoning and the Roman numerals, and fol- 
lowed these by the Hindu arithmetic. 5 The teaching of 

1 Cardan, Practica Aritfometice (1539), Chapter II, gives seven: numer- 
ation, addition, subtraction, multiplication, division, progression, and the 
extraction of roots, 

2 Sigismund Suevus, Arithmetice Historica (1593), fol. aii recto. 
Gemma Frisius, Arithmeticae Praoticae Methodus Facilis (1575 ed.). 

" Solent nonnulli Duplationem & Mediationem assignare species distinctas 
a Multiplicatione & Divisione. Quid vero mouerit stupidos illos nescio, 
eu & finitio & operatio eadem sit." Fol. B 5 verso. 

3 Unger, Die Methodik, page 72 f § 41. 

4 A few writers included the Rule of Three — Riese for example. 

5 Kobel, Zwey reohenbuchlin (1537 ed.). After teaching the Roman 

symbols, I, V, X, L, C, D, M, and the digits, 1, 2, 3, 9, Kobel gives 

a comparative table entitled : " Tafel zu erkennen vnd vergleichen die zal 
der Buchstabn aufi dem A. b. c. genomen/ vnd der Figuren/ die man 
ziferen nennet/ Underrichtung." Fol. B 6 recto and verso, or 14 recto 
and verso. 

In the table Kobel uses small letters for all numbers except ten and 

multiples of ten, as i, ij, iij, iiij, v, - - X, xj, xij, - - XX . For 500 

he uses D«; for 1000, jM; 2000, ijM; for 100 he gives both C >aind j c ; 



38 



SIXTEENTH CENTURY ARITHMETIC 



numeration was a formidable task, since the new notation was 
so unfamiliar to the people generally. The feeling was prev- 
alent that one must learn the Roman system and then graft on 
the new system. 

Numbers in the Hindu system were divided into three 
classes: (i) digits (digiti), 1,2, — ,9; (2) articles (articuli), 
ten and multiples of ten, 10, 20, 30, — ; (3) composites (com- 
positi), combinations of digits and articles, as 25, 37. Most 
writers stated the idea of place-value simply and directly, but 
some, owing to its novelty, gave it an elaborate treatment. 1 

The names of the orders and the device used as a separatrix 
varied extensively. The names to hundred thousands were 
the same as now used, but the period now called millions was 
usually called thousand thousand. 2 The word million dates 

200, cc and ij c ; 1100, fylC, also Mj c ; 1200, Mcc, also Mijc; for 1300, 
Mccc, also Miijc. 

1 Wi'dman, Behend und hiipsch Rechnung (1508 edi), fol. 6 recto. 
Robert Recorde, The Ground of Arites (1594 ed.). Recorde, whose 

book is in dialogue form, thus quaintly develops the idea of place value : 

" But here must you marke that everie figure hath two values : One 
alwayes certaine that it fignifieth properly, which it hath of his forme: 
and the other vn certaine, which he taketh of his place." Fol. Cvi verso. 

"M. (Master). Now then take heede, thefe certaine values euery figure 
reprefenteth, when it is alone written without other figures joyned to him. 
And alio when it is in the firfte place, though manie other <io follow ; as 
for example: This figure 9 is ix. standing notw alone. 

" Sc. (Scholar). How: is he alone and ftandeth in the middle of fo 
many letters? 

"M. The letters are none of his felowes. For if you were in France in 
the middle of a M. Frenchman, if there were none Englifh man with you, 
you would reckon your felfe to bee alone. 

" Sc. I perceiue that. And doeth not 7 that standeth in the second 
place betoken vii? and 6 in the third place betoken vi? And so 3 in the 
fourth place betoken three ? 

" M. Their places be as you haue laid, but their values are not To. 
For, as in the first place, euery figure betokeneth his owne value certaine 
onely, fo in the second place euerie figure betokeneth his owne value a 

hundretn times, fo that 6 in that place betokeneth vi. C. ." Fol. Cvii 

verso. 

2 Raets, Een niew Cijfferboeck/ (1580). " Duyfentich duyfent 

ioooooo." Fol. Aiij recto. 



THE ESSENTIAL FEATURES 39 

back to the thirteenth century; * but the sixteenth century 
records the struggle of the word for its place in numeration. 
Borgi 2 (1484) has it in "Million de million de million," 
Chuquet ( 1484) used it in reading numbers on the six-figure 
basis. 3 Paciuolo 4 (1494) used " milione." Cirvelo 6 (1495) 
used million for 1,000,000,000,000. La Roche (1520), like 
Chuquet, used it on the six-figure plan. After 1540 the word 
appeared in many standard works. 6 The present names for 
higher periods, though much slower to come into use, were 
known to fifteenth century scholars. Chuquet (1484) gave 
the remarkable list : " byllion, tryllion, quadrillion, quyllion, 
sixlion, septyllion, ottyllion, nonyllion," using them on the 
six-figure basis. La Roche (1520) gave billion and trillion. 

Tonstall, De Arte Supputandi (1522). Tonstall reads in Latin the num- 
ber 3210987654321 as Ter millies millena millia rnillies, ducenties decies 
millies millena millia, noningenties octvagies septies millena millia, sexcenta 
quinquinta quattuor millia, trecenta viginti unum. Fol. C 2 verso. 

1 The word "million" is first found in Marco Polo (1254-1324). 

2 The 1540 edition of Borgi's work gives the following names of high 
periods: " Miar de million, million de million, miar de millio de million, 
million de million de million." Fol. 5 verso. 

The word is also used in the Treviso arithmetic (1478). 

3 By the six-figure basis is meant the use of millions to cover six orders 
beyond hundred thousands, billions to cover the next six orders, and so on. 

Thus, 18,432,750,198,246,115 would be read eighteen thousand four hundred 
thirty-two billion, seven hundred fifty thousand one hundred ninety-eight 
million, two 'hundred forty-six thousand one hundred fifteen; instead of 
eighteen quadrillion, four hundred thirty-two trillion, seven hundred fifty 
billion, and so on, on the three-figure plan. The six-figure grouping of 
Chuquet and La Roche entered Germany in 1681, according to Unger (Die 

Methodik , p. 71) and came into general use there in the eighteenth 

century. France early adopted the three-figure system. England used the 
old terminology at the opening of the sixteenth century, for Tonstall (1522) 
says that millena millia (thousand thousand) is commonly called "mil- 
lion " by foreigners. But before the middle of the century we find Recorde 
using million. 

4 Paciuolo, Suma (1494 ed.), fol. 9 verso. 

5 Cirvelo, Tractatus Arithmetice practice (1505), uses the same notation. 

6 Recorde, The Ground of Artes (1540) ; Gemma Frisius (1552, Antwerp 
ed.) ; Cataldi (1602 ed.). 



4 SIXTEENTH CENTURY ARITHMETIC 

Van der Schuere (1600) gave a very large number field. He 
used millioen for million, duyset mill, (thousand million) for 
billion, bimillioen for trillion, duyset bimill. for quadrillion, 
and so on up to duyset quadrimill. for octillion. Trenchant 
(1578) gave millions for million, miliar for billion, and milier 
de miliars for trillion. 

The following devices were used for separating the periods : 

678935784105296 * 5678900000000000000 2 3210987654321 3 



foacbacba 4 :: :: .. .. 

44559886 3 J 554 1 560 5 23456007840000305321 6 

1 . 234 . 567 . 890 7 36236365463643656765656568 8 

4 3 2 1 

Recorde (1540) called the numbers in each period ternaries 
and the periods denominations to assist in reading. Thus, as 
in 222 pounds, pounds is the denomination, so in every period 
{620,000) the last place (thousand) is the denomination. 

1 Leonardo of Pisa, Liber Abaci (1202, or 1228), p. 1. 

2 Borgi, Qui comeza la nobel opera (1540 ed.), fol. Av verso; Paeiuolo, 
Suma (1523 ed.), fol. 19 verso, oiii verso. 

3 Ton stalU, De Arte Supputandi (1522), fol. C 2 verso; Kobel, Zwey 
Rechenbuchlin (1537 ed.), fol. B 4 verso; Rudolff, Kunstliche rechnung 
{1534 ed.), Aiij recto; Riese, Rechnung auff der Linien unid Federn/ 
(1571 ed.), fol. Aij verso; Baker, The Well Spring of Sciences (1580 ed.), 
fol. Biiii recto. 

4 Kobel, Zwey Rechenbuchlin (1537 ed.), fol. B 5 recto. 

5 Gemma Frisius, Arithmeticae Practicae Methodus (1581 ed.), fol. Aiv. 

6 Tartaglia, La Prima Parte (1556). 

7 Ramus, Arithmeticae Libri Duo (1577 ed.), fol. Aii verso. 

8 Unicorn, De L'Arithmetica vniversale (1598), reads this number 
thus: 36. millionii quatro volte, & ducento trenta sei millia, & trecen- 
tosessanta cinque millioni tre volte, et quatro cento e sessanta tre millia 
<S* sei cento e quaranta tre millioni due volte, & sei cento cinquanta sei 
millia e setteoento sessanta cinque millioni vna volta, & sei cento e cinquata 
sei millia, e cinquecento sessanta otto. Fol. A 4 recto. 



THE ESSENTIAL FEATURES 



41 



The number ini the period he called the numerator. Thus, in 
203,000,000, 203 is the numerator. 1 

Addition 

The treatment of addition presents much diversity, but the 
general characteristics are the absence ®i tables of sums, full 
explanations of column-adding, and tests of the work. 

It would seem that the sums corresponding to the modern 
addition table would necessarily have received first attention 
at a time when the Hindu numerals were so unfamiliar. But 
the writers who used these were the exception. 2 

The explanation- of the processes of column-adding usually 

1 Recorde, The Ground of Artes (1540). 
Scholer. What call you Denominations ? 

Mafter. It is the lafte value or name added to any fumme. As when 
I fay: GCxxii. poundes: poundes is the Denomination. And iikewife in 
faying: 25 men, men is the Denomination, and fo of other. But in this 
place (that I fpake of before) the laft number of euery Ternarie, is the 
Denomination of it. As of the firft Ternarie, the Denomination is Unites, 
and of the feconde Ternarie, the Denomination is thoufandes: and of 
the third Ternaries, thoufande thoufandes, or Millions: of the iiii, thou- 
fande thoufande, thoufandes, or thoufande Millions : and fo foorth. 

Scholer. And what fhall I call the value of the three figures that may 
be pronounced before the Denominators : as in faying 203000000, that is 
CCiii. millions. I perceyue by your wordes, that millions is the denomina- 
tion : but what fhal I call the CCiii. joyned before the millions. 

Mafter. That is called the Numerator or valuer, and the whole fumme 
that refulteth of them both 1 is called the fumme, value or number. Fol. 
Dii recto (1594 ed.). 

2 Tartaglia, La Prima Parte Del General Trattato (1556). Tartaglia 
gives the addition tables as follows : 

o. e 0. fa (0 + = 0) 
o. e 1. fa 1 (0 + 1 = 1) 
o. e 2. fa 2 (0 + 2 = 2) 



o. e 10. fa 10 



1. e 1. 


fa 2 




1. e 10. 


fa 11 


and so on with all the 
.tables to 
10. e 10. fa 20 


2. e 2. 


fa 4 


2. e 10. 


fa 12 


Fol. B i verso. 



That these tables were regarded as fundamental to further progress is 
shown by the following remark from the siame folio : " Imparate adunque 
li soprascritti sumari necessarij di saper a menti." (Therefore to learn to 
add, it is necessary to commit the tables written above to memory.) 



4 2 SIXTEENTH CENTURY ARITHMETIC 

received first attention. Two abstract numbers were proposed 
for addition, such that the sum of one column of figures, at 
least, would equal or exceed ten. The first example in Een 
niew Cijfferboeck, by Willem Raets (1580) is: 

354 



1252 
Cardan's (1539) first example is: 

73942 

4068 

273 

52759 



131042 



Noviomagus (1539) first shows the arrangement by columns, 
as in (1) ; then adds without carrying, as in (2) ; then adds 
with carrying, as in (3) : 



(I) 


(2) 


(3) 


321, not 321 


321 


2354 


124 124 


421 * 


620 


530 530 


530 


76 



975 3050 



Tonstall (1522) gives: 



[I) 


(2) 


(3) 


(4) 


4 


309 


59 


389 


3 


204 


34 


204 



7 513 93 93 

He points out in example (2) that the ten in the sum of the 
first column falls under the second column, because the second 
column is composed of zeros. Examples (3) and (4) are ex- 
plained in order, although it would seem that (4) should pre- 
cede (3). The student is advised by Tonstall to learn the ele- 

1 Typographical error in the original for 124. 



THE ESSENTIAL FEATURES 



43 



mentary sums, and is encouraged by the remark that it will 
require only an hour. 

It was usual in addition to arrange the addends in order of 
size, placing the largest at the top. It is easy to state a reason 
for this, although none is given; for by this plan the columns 
were more easily preserved — a real difficulty for beginners, 
especially for those to whom the Hindu system: was unfamiliar. 
It also prepared for subtraction, but this was probably not 
the reason for using it, as is shown by 

by the following examples : * Trie 3456 4602 56789 
first is the only one that corresponds 9 ° 54 34 ° 5 2345 

1 ■ U^ «J ■ > 2300 I234S 

to an example in subtraction, since it 6 7 g 9 

has only two addends, but here the 10307 3239 

smaller is written at the top. ^^ 

1327 
234 In the other examples, which do not corres- 6 

345 pond to those of subtraction, the numbers 349 

5 are in the order of their size from the top 228 

763 down. The illustration at the left shows 3 ^ 

832 how arrangement according to size was 3 8 

3450 occasionally disregarded. 2 

63 The sums of the several columns in a * 32 
13 problem of addition were commonly added 

as at present, by writing the first right-hand figure of 
1000 ^ e suro °f an y column and adding the rest to the next 

column. When the columns are long, how- 

7825 e ver, -tkjg j s not the eas i es t wa y, and occa- j^ 3 

sionally a writer of that period wrote the g786 

partial sums and added them to obtain the result. 3 

A slight modification of this was the placing of the J 4 2 9o 
numbers to be added to the next column below that 



column to be added to its sum. 4 ^ 5 

A feature that is written large in arithmetic of the six- 

1 Rudolff, Kumstliche rechnung (1534 ed'.), fol. Aidij recto. 

2 Tagliente, Libro Dabaoo (1541 ed.), fol. Ci recto. 

3 Gemma Frisius, Arithmeticae Practicae Met»hodus facilis (1575 ed.), fol. 
Aviii verso. 

4 Buteo, Logistica (i559), fol. a 7 verso. 5 Error in the original. 



44 



SIXTEENTH CENTURY ARITHMETIC 



teenth century is the matter of the so-called proofs of opera- 
tions. These were generally not proofs, but tests more or less 
reliable. The most common form was that of 
casting out nines. Thus, in the annexed case 
of addition, the remainder arising from divid- 
ing 354 by 9 is 3, and from dividing 898 by 1252 
9 is 7. The excess of nines in 7 + 3 is 1 ; 
this is the 1 above the line in t- The excess of nines in 1252 
is 1 ; this is the 1 below the line, t shows that the excesses 
agree and that the work checks, or proves, as it was called. 
It was customary to give a long explanation of the proof, and 
a few writers gave a table of remainders arising from divid- 
ing numbers from o to 90 by 9, and showed how to find the 
remainders for large numbers. 2 

That tests were given exaggerated importance is shown by 
the fact that several writers extended them to the case of add- 
ing denominate numbers. The excess of nines was found for 
the highest denomination; this was expressed in terms of the 
next lower denomination and combined with it. Then the 
process of finding the excess was repeated. Each addend and 
the result were similarly treated, the work of testing becoming 
more difficult and complicated than the solution of the problem. 

The test by casting out sevens was also common, but being 

1 Raets, Eem niew Cijffierboeck (1580), foil. Aiiij ireotb. 

2 Tartaglia, La Prima Platte (1556), fbl. Bij verso and fol. Biij recto. 

Li termini della proua del. 9. 

De 0. la proua e De 10. la proua e 1 

De 9. la proua e o De 11. la proua e 2 

De 18. la proua e o De 12. la proua e 3 

De 90. la proua e De 19. la proua e 1 

De o. la proua e De 21. la proua e 3 

De 1. la proua e 1 De 22. la proua e 4 



De 2. la proua e 2 

De 30. la proua e 

De 9. ia proua e o, and so on to: 

De 81. la proua e 



De 90. la proua e 0. 



THE ESSENTIAL FEATURES 45 

more difficult, it was placed second. Several authors mention 
the fact that it is more accurate than that by casting out nines. 1 
The proof by elevens was sometimes used. In addition, an 
author occasionally used the method of adding the columns of 
figures both upward and downward. 2 A few used subtrac- 
tion, 3 in the case of two addends taking one addend from the 
sum to see if the result is the other. The use of subtraction 
in the case of more than two addends was rare. 4 The reason 

1 Unicorn, De L'Arithtmetica vniuersale (1598), "Che la proua 'del 7. sia 
men fallace, die la proua del 9," fol. B 2 verso. 

2 Paoiuolo, Suma (1523 ed.), fol. 20 recto, or Ciiij recto. 

3 Taritaglia, La Prima Parte (1556), gives this example: 8756 

a 678 b 

9434 
and says : " Perclhe inuero il sommare e proprio in atto contrario al sot- 
tare, & similmente il sottrare e in atto contrario al sommare," fol. Bij 
recto (because, indeed, addition is properly the inverse of subtraction, and 
similarly subtraction is properly the inverse of addition). 

4 Champenois, Les Institutions De L'Arithmetique (1578). 

4325 Preuue. 
132 4878 fome de 1' Addition. 
421 4325 premiere fomme. 

Addition 4878 0553 premier refte. 

132 feconde fomme. Page 15, or fol. Bviii recto. 

421 fecond refte. 

421 troifiefme fomme. 



000 

Also Simon Jacob, Rechenbuch auf den Linien undmit Ziffern/ (1599 ed.). 
597- a 
786.b 
978.C 

Summa 2361 
597-a 

1764 Fol. Cv recto. 

786.b 

978 
978.C 

000 



4 6 SIXTEENTH CENTURY ARITHMETIC 

for the prominence of these tests is undoubtedly due to the use 
of the various forms of the abacus. When the beads were 
once shifted, or the counters displaced, or the symbols in the 
sand effaced, there was no record to retrace, no possibility of 
reviewing the work. 1 It was, therefore, very advantageous to 
have means of testing the result by some comparison with 
the original numbers. These means were supplied by the 
proofs of nines and sevens. It was natural, then, that these 
tests should appear with due emphasis in most of the first 
printed books. 

Besides the general characteristics of sixteenth century ad- 
dition there were a few special features that have educational 
significance; namely, the order of adding, certain short meth- 
ods, and the use of concrete problems to introduce the process. 

We have noted that Paciuoloi added upward and then added 
downward as a test of the work, but only one writer among 
those examined confined his addition to the downward pro- 
cess. 2 Thus, whatever virtue there may be in precedent is in 
favor of adding upward instead of downward. 

1 For some time after arithmeticians formed the habit of writing num- 
bers in the Hindu notation, they used line-recfeoniing to perform the pro- 
cesses, and to watch their progress they crossed the figures as they were 
used. The influence of this is seen in the following example from Kobel 
(Zwey Rechenbuchliin (1537 ed.), fol. in verso, 112 recto), in which 
he crossed the figures of the addends, although using the Hindu algorism : 

Zum erf ten wil ich zufammen thun 103 zu 966. - - - Ich 

fprich/ 6. vnd 3. ist 9. vnd fetz die 9 vnder das ftrichlin vff die 102 

erft flat/ vnd durehftreich 'die 6. vnd 3. fo fteht es alio. 6 * 

ftetz ich vnd den ftrich/ vff die zweyte ft-at neben die 9. zu der 9 
lincke hand vnder die 6. vnnd durcbstreichs o. vnd 6. 

w 

m 

69 

2 Trenchant, L'Arithimetique (1578 ed.). 

"Je veux aiouter ces nombres, 581, 192, & 264. - - - Commencant a 
main droite, i'aioute toutes les figures du denier reng enfemble, 
diftant, 1 & 2 font 3, & 4 font 7. ie pofe 7 fous celuy reng an 581 

deffous du tret, & vien femblablement cueillir e precedent reng, 192 

diftant 8 & 9 font 17, & 6 font 23, ie pofe le digite 3 fous ce 264 

reng, & retien le nombre des dizeines qui eft 2, que i'adioute auec [1037 
d'autre reng, diftant, 2 que te tien & 5 font 7, & 1 font 8, & 2 
font io, ie pofe o & retien 1, que ie pofe deuant o, & c'eft fet. Ainfi ces 
troys nombres aioutez montent 1037. Fol. Biv verso. 



THE ESSENTIAL FEATURES 47 

Another feature of sixteenth century calculation that one 
would not expect so early in the history of European figure- 
reckoning was the use of short methods. They were not com- 
monly used, but there is a fair sprinkling of them through the 
various operations. A few writers showed how equal num- 
bers are combined while adding a column. 1 A few cases occur 
in which the associative law is used to break up a long prob- 
lem into shorter ones. 2 

1 Reoorde, The Ground of Artes (1594 ed.) : "I would) 4889 
adde thefe xiii fumes into one, which I let after this manner: 4599 
then doe I begin and gather the fumme of the firft rowe of fig- 2299 
ures which commeth to 107, for I take 9 there x. times and that 3699 
is 90, then 9 and 8 is 17, that is in all 107, of which fumme I 2399 
write the 7 under the firft rowe of figures, and then for that 4090 
100 is x. tens, I keepe x. in mind: which ten I muft adde vnto 1099 
the nexte rowe of figures when they are added together with the 3198 
x. that I had in my minde, make in all 125, of which fumme, I 299 
write the digit 5 vnder the fecond rowe, z . Then for that 120 699 
conteineth xii tens. Eol. Cii recto. 899 

Cirvelo, Tractate Arithmetice practice (1513 ed.)- 499 

"Et nota qj ad iftam fpeciem reducitur alia fpecies minus 389 

principalis que dioitur .duplatio aut triplatio nam fi eundem 

numerum bis fcripferis et addideris in vnam fummam habebis 29057 
duplum illius : ynde pro re tarn f acili no oportebat dare fpeciale 
capitulum. Exemplum." Fol. aiiij verso. 

496 2 496 3 

496 2 Exemplu 496 

496 3 

992 -duplum 

1488 triplu 
An application of doubling and tripling to addition. 

2 E. g., Clavius, Arithmetica Prattica (1626 ed.), fol. a 4 recto. 



6008 


308 IO8 3009 


6008 


5009 


239 309 209 


5009 


4009 


108 4128 308 


4009 







308 




15026 


655 4545 3526 


239 
108 
108 


15026 




309 


655 


The top row furnishes partial sums 


4128 


4545 


of the column at the right. 


3009 


3526 




209 
308 


-9"3'7C"2 



23752 



48 SIXTEENTH CENTURY ARITHMETIC 

The plan of proposing a concrete problem in addition as a 
motive for explaining the process occurs in several works. 
The following will serve to illustrate : 

"As if there were due to any man 223 pounds by some one 
body, and 334 pounds by another, and 431 by another, and 
you would know how many pounds is due to the same man 
in all." x 

" A merchant has three purses in which there is a certain 
number of ecus. There are known to be 3,231 ecus in the 
first, 2,312 in the second, and 1,213 m tne third. The mer- 
chant put the contents of these purses into one. It is required 
to know how many ecus there are in this purse." 2 

" For example, if it is asked how long ago Homer lived, 
and Gellius replies: 160 years before the founding of Rome, 
which was founded 752 years before the birth of Christ. 
Christ was born, however, 1,567 years ago. These three num- 
bers are added. The sum showing that Homer flourished 

2,479 years ago will be as follows " : 3 160 

752 
1567 

2479 

It is somewhat remarkable that the idea of introducing a 
process through concrete examples should have taken root in 
so many countries; within a period of fifty years at a time 
when communication of ideas was so slow. Kobel in Ger- 

1 Baker, The Well Spiring of Sciences (1580 ed.), foil. Bvi recto. 

2 Ghampenois, Institutions De L'Arithmetique (1578). 

"Vn marchant a trois bourfes ou il y a certaines fommes d'efeus, 
fcauoir en la premiere 3231 efcus, en la feconde 2312 efcus, & en la troi- 
fiefme 1213 efcus. 

Ce imarchat vuide fes trois 'bourses en vne. Lon demande combien i'l y 
a id'efcus en cefte bourfe." Fol. Biiij recto, or page 7. 

3 Ramus, Arithmebicae Libri Duo (1577 edl). 

" Ut f i quaeratur quampridem vixerit Homerus, & respondeatur e Gellio, 
160 annis ante conditam Roman, quae condita lit ante natum Chriftum 
annis 752. C'hriftum vero natum anno abhinc 1567. addaritur hi tres 
numeri : Summa inductionis indicans Homerum annos abhinc 2479 floruif fe, 
erit hoc modo." Fol. Aiiij recto. 



THE ESSENTIAL FEATURES 49 

many (1531), Recorde (1540) and Baker (1562) in Eng- 
land, Ramus (1567), Trenchant (1571) and Champenois 
(1578) in France were the pioneers in their respective coun- 
tries. Many who began with abstract numbers introduced de- 
nominate numbers after the first two or three problems. 

Subtraction x 

Another evidence that this was the formative period in ele- 
mentary arithmetic is seen in the treatment o>f the subtraction 
of integers, for these writers were in possession of all the 
methods of subtraction that are taught or discussed at the 
present time. 

There was little variation in their treatment where the fig- 
ures of the minuend had greater value than the corresponding 
ones of the subtrahend. In fact, all the writers included in 
this investigation, with one exception, subtracted from right to 
left the figures of the subtrahend from the figures oi the min- 
uend written above, and placed the differences below the cor- 
responding columns. 

1 A knowledge of the elementary differences required for this was pre- 
supposed, although the tables were given in the more elaborate works only. 
Ramus (1586 ed.) recommended learning the " alphabetum " both for addi- 
tion and subtraction. By alphabetum he meant all the possible sums and 

differences of the digits, 1 9. " Subductionis mediatio in primis novem 

notis eadem hie effe debet, quae fuit in additione. 

"Tolle 3 de 7 manent 4, tolle 4 de 9 manent 5. & fimiliter totu alpha- 
betum 1, 2, 3, 4, 5, 6, 7, 8, 9. Omni genere verfandum est. Hie Pytha- 
goreus fubductionis abacus eft." Fol. a 4 verso. 

Tonstall (1522), after explaining how to subtract numbers of several 

figures, states in words the differences from 1 9 and recommends that 

they be learned. 

" Quod f i quis ignorat : unius horae labor ; 
Modo intentus fit animus ; if suppeditabit." Fol. F ± recto. 



5° 



SIXTEENTH CENTURY ARITHMETIC 



Ramus began at the left and proceeded to the right. 1 
This is his first example, where the influence of line- 
reckoning is again seen in the crossing of the figures. 
Ramus was not the first to subtract from left to right, for 

The following is TonstaH's table: 



111 



m 



19 


18 


17 


16 


15 


14 


.,,.13 u 


12 


11 


1 19 


i 13 


' 17 


16 


15 


14 


IS 


IS 


11 


9 1 13. 


J2. 


_a 


J& 


_a 


»a 


-& 


«a 


_g 


\ 9 


9 . 


- s 


7 


ft 


s 


4- 


R 


s 






17 


16 


15 


14 


13 


12 


12 




8 


JBL 


_a 


.a 


_a 


-a 


ja 


-J 






a . , 


R 


•? 


R 


R 


4 


s 






16 


15 


14 


13 


12 


11 




7 




JL 


JL 


JL 


-1 


JL 


-3{ 








— a— 




f 


fl • 


5 


_— i 






15 


14 


13 


12 


11 






e 




-£ 


_£ 


-a 


-A 


-5 










« 


ft 


„., v 


fl 


fi 






14 


13 


12 


11 






5 






9 


-* 


-5 

7 








13 


12 


11 








4 






9 


-4 


7 






12 


11 
















-A 


Ji 












3 




fl 


R 






11 














z 




-1 



Folio F 2 recto. 
The following is Tartaglia's (1556 ed.) table: 

De 0. a cauarne o. resta o De 2. a cauarne 2. resta — 

De 1. a cauarne o. resta 1 De 3. a cauarne 2. resta — 



De 10. a cauarne 0. resta 


10 


De 10. a cauarne 2. resta • 

De 3. a cauarne 3. resta - 
De 4. a cauarne 3. resta - 


8 


De 1. a cauarne 1. resta - 
De 2. a cauarne 1. resta - 


— 

— 1 




— 1 



De 10. a cauarne I. resta 9 De 10. a cauarne 3. resta 7, 

and so on to the table 10 — 10 = o, which contains this one fact only. 

Fol. C ± recto. 

1 Ramus, Arithmeticae Libri Duo (1586 ed.). "Si dati lint plurium nota- 
rum fubducendo infra alterum pofito, fubductio fit a finiftra dextrorfum, 
reliquoq? ; f upernotaito delentur dati, ut f i die f umma aeris illius alieni 345 
fubduceda fit 234 : Dispofitis ordine numeris hoc mo-do : 345 fubducendo 
infra, fupra autem a quo fubductio faciendla, incipiam a 234 
finiftra dextrorfum, contra qua in additione, tollo 2 de 3 manet 1, & 
fupernoto 1 deletis 3 & 2. Deinde fubducam 3 de 4 manet 1, & 
fupernoto 1 deletis 4 & 3. Deniq? fubductis 4 de 5 manet 1, & 
fupernotabo 1 deletis 5 & 4. Vnde inveniam reliquu effe III. cum 
fubduxero 234 a 345. Inductio tota fie erit. Fol. A 4 verso. 



Ill 

m 

m 



THE ESSENTIAL FEATURES 5 1 

the same thing was done in the Lilivati, and possibly in older 
works. 1 Many calculators to-day recommend working from 
left to right in both addition and subtraction, usually confining 
the addition to two addends. But Ramus added from right to 
left in all cases and subtracted from left to right. He was 
unique in his century, also, in placing the difference above the 
minuend, as shown in the example. 

The case of subtraction in which the subtrahend figure ex- 
ceeds in value the minuend figure, received a diversity of treat- 
ment. The three distinct methods usually taught were: (1) 
Ten is added to the minuend figure before the subtrahend 
figure is subtracted ; one is then added to the next subtrahend 
figure. (2) The arithmetic complement of the subtrahend 
figure is added to the minuend figure, and one is added to the 
next subtrahend figure. (3) Ten is added to the minuend 
figure, and the subtrahend figure is subtracted from this sum ; 
one is then subtracted from the next minuend figure. The 
last is the form of solution most prevalent to-day. Lists of 
authors who used these respective methods are given below ; 2 
from these lists it will be noticed that the first and second 
methods were equally popular, while the third method was 
used very little. Ramus, subtracting from left to right, used 
the third method, for which he gave the following example 

1 See H. Suter, Bibliotheca mathematica, VTI 3 15. Gerhardt, "Etudes," 
page 5. 

2 Those who used the first kind were: Piero Borgi (1484), Tonstall 
(1522), Paciuolo (1494), Rudolfr" (1526), Cardan (1539), Noviomagus 
(iS39), Tartaglia (1556), Van der Scheure (1600). Ton- 
stall also gave this process in the following form: 2 9 10 10 

Those who used the second ford were: Widman (1489), p V K v 
Tonstall (1522), Paciuolo (1494), Tartaglia (1556), Gemma 1111 
Frisius (1540), Riese (1522), Trenchant (1571), Baker 18 9 9 
(1562), Unicorn (1598), Huswirt (1501), and Finaeus 
(1525). This method goes back to the Hindu arithmeticians. Fink, 
Geschichte der Elementar-Math. (Beman and Smith's translation, Chicago, 
1900), p. 28. 

Those who used tthe third kind were: Paciuolo (1494), Kobel (1531), 
Tartaglia (1556), Champenois (1578), Buteo (1559), and Raets (1580). 



52 



SIXTEENTH CENTURY ARITHMETIC 



and explanations: "When I take 3 from 4, I shall not 87 
write 1, because the following subtrahend figure, 4, is £?£ 
greater than the 3 placed above, but I shall keep this in ^ 
mind and take the next figure below, which is 4, from 13. 
This leaves 9, which for the same reason I shall not write 
down, but shall take 1 from it and write 8 above, and keep 1 
in mind, because the following figure to be subtracted is 
greater; then 5 from 12 leaves 7, which will be written 
above." * 

The proofs in subtraction, as in the case of all operations, 
were very prominent. The three standard methods were cast- 
ing out nines, casting out sevens, and adding the subtrahend 
and difference. 2 About three times as many writers used the 
additive method as used either of the others, which was nat- 
ural on account of its ease and effectiveness. 3 Tartaglia, who 
gave each of the proofs above, also subtracted the remainder 

1 Ramus, Arithmeticae Libra Duo (1586 ed.). "Vt fi fubduoenda finit 
345 de 432, cum fubdlucaon 3 de 4 non fuperinotabo 1, quia 4 fequenis 
fubducenda note major est fuperapofita 3, fed illud .mente refervabo, 4. 
fubductis a 13 maneret 9, quae nequaquam propter eandem caufani notabo, 
fed uno minus 8 tantum fupernotabo, & unum mente refervabo, quia, 
fequens fubduceda nota major est. Itaq? 5 rubductis a 12 reliqua 7 fuper- 
notabo. Vnde inveniam fubductis 345 de 432 relinqui 87. Tota inductio 
fie erit : 87 

m 
m 

"Trium foiciorum pecunia in unum acervum congefta fit 432: primiqj 
fumma fit incerta, conftet tamen focios capere 345: ergo fuam partem is 
per hac fubductionem cognof cet." Fol. A 5 recto. 

2 Piero Borgi, Arithmetica (1540 ed.), fol. C 6 verso: Example. Proof. 

456 333 
123 123 

333 456 

3 The proof by casting out nines was used by Rudolff (1526), Widman 
(1489), Cardan (1539), Tartaglia (1556), 'Gemma Frisius (1540), Suevus 

(1593). 

The proof by casting out sevens was used by Widman (1489), Cardan 
(1539), Tartaglia (1556), Unicorn (1598). 

The addition proof was used by Borgi (1484), Widman (1489), Cardan 
(1539), Tartaglia (1556), Riese (1522), Champenoiis (1578), Raets (1580),, 
Unicorn (1598), Jacob (1599), Van der Scheure (1600). 



THE ESSENTIAL FEATURES 



53 



from the minuend to find the subtrahend, as in addition he 
subtracted the sum of all the addends but one from the result 
to find the other addend. 

The terminology and symbolism used have several points of 
interest. In those works which used the plan of supplying 10 
from the next order of the minuend to make subtraction pos- 
sible, one naturally seeks to find 
a trace of the modern vulgarism 
f to borrow," and recognizes it in 
the word " entlehen " used by 
Kobel. 1 This is suggestive, be- 
cause Kdbel was primarily an aba- 
cist, and he would probably em- 
ploy the same word in the algor- 
ism that was employed to describe 
the actual borrowing process in 
abacus reckoning. In the tables 
and examples of subtraction there 
is suggested the word " rest " in 
the sense of difference, or re- 
mainder. 2 The book material of 
addition and subtraction presents 



4 « > fen ot»cr t>e{?glcy* 

> -f* So ctoerr/Sofumier 

4 '? btegenttnercnb 

3 H~ 44 ttxmnt) waeauft 
3 + ii — ifi/Oaetfiim* 

Smtnet 3 1 1 K> nu6t>$fa$cfon* 

3 + f fcervnnb werbeit 

4 «* 47J9& C&O 

3+44 t>u bi'e jenbtncr 

3 4* a 9 Jn&gemacbete 

3 — *— »i a (?afienrtbt>4S / 

5 4- 9 <-f &<*6 i(i meet 

OAr,Q2t>l>tcrcft>iiO>7 minus. Hun 

folcfcuftfr §$l%ab\~d)lal)zn allweea for 

ainlegdi4K,.>c>n&&a8tfl 1 3 malz4. 

Wtt>mad)C3 1 z tt&arjua&btcrD.io — 

t(Jdi(J>5tt>enbi»erv)cn3SJ' s^ye(uS« 

crazier »dn4<T3 ^.TCnD fycybcn 4 » <fz 

tt.njmfpjid) j 00 tt>t>a&i{teim>entnet 

|«04 ff i tvtc Pnnlcn 4 1 7 i tfc cn& fuml 



W«fftt 



1 Kobel, Zwey Rechenbuohlin (1537 ed.), fol. P 4 recto et seq. 

For discussion see Brooks, Philosophy of Arithmetic (1901 ed.), PP- 45, 
46; 219, 220. Unger, Die Methoidik, pp. j$, 74. 

2 Tartaglia, La Prima Parte (1556). " Sottrare non e altro che duoi 
proposti numeri, inequali saper trouare la loro differentia, cioe quanto che 
il maggiore eccede il menore, come saria a sottrare. .4. de .9. restaria .5." 
Fol. Bvi verso. 

79374 
5024 



74350 II numero restante. 

Unicorn, De UAnithmetica vmiversale (1598), calls the remainder numero 
restante, or resta dare, or. residue. 

Trenchant, L'Arithmetique (1578 ed.), calls the remainder reste. Fol. 
B g recto. 

Baker, The Well Spring of Sciences (1580 ed.), does not use the terms 
minuend and subtrahend, but says, in taking 6 and 9, "there resteth 3." 
Fol. Ciii verso. 



54 



SIXTEENTH CENTURY ARITHMETIC 



a much different appearance from that of the modern treat- 
ment, because of the lack of symbols of operation. Although 
the symbols -f- and — were in existence in the fifteenth cen- 
tury, 1 and appeared for the first time in print in Widman 2 
( 1489) , as shown in the illustration (p. 53) , they do not appear 
in the arithmetics as signs of operation until the latter part of 
the sixteenth century. In fact, they did not pass from algebra 
to general use in arithmetic until the nineteenth century. They 
were used in the sixteenth century to express excess and deficit 
in weight, 3 as shown in note 3, where the first column is sent- 
nets and the second pounds. These early printed examples 
substantiate the theory that the symbols +, — , originated 

1 Miiller, Historisich-Etymologische Studiien iiber mathematische Termin- 
ologie. 

2 Widman, Behend und hupsch Reohnung uff alien Kauffmanschafften. 
(1508 ed.), fol. h 5 recto. 

3 Weiracelaus, T'Fondament Van Arithmetiea (1599 ed.). 

" Pijpen Olie van Oliven/ weghende alsoo -hier naer volcht/ Tara op 

elcke pijpe/ 140. Ib.i. lauther 100. cost 50. H. 6. <d. Hoe veel beloopter in 

ghelde?" 

Weecht. 

cent. lb. 

No. 4. 9 + 50. 

No. 5. 9 + 55- 

No. 6. 9 + 56. 

" Dit + beteyckent Plus/ ende dit — Minus." 

" Item 9. pipes, d'huyle d'Olives pesants eomme s'ensuit tara pour chas- 
cune pipe 140. lb. & couste 1. cent netto 50 H. 6 d. Combien monte le tout 
en argent. 

"Facit L. 186. fl. 4. d. io 14 / 25 . 

" Cecy -f- signifie plus, cecy — moins." Page 59. 

The problem is the same in each case, the book being printed in two 
languages in parallel columns. The translation of the problem is: 

" Nine casks of olive oil have the weights given below, the tare for each 

cask is 140 lb., and 1 centner net costs 50 florins 6 denarii. What is the 

entire cost?" 

Weight. 







cent. 


lb. 


No, 


1. 


9 + 38- 


No. 


2. 


9 + 


44. 


No. 


3- 


10 — 


20. 







ceni 


l . lb. 


No. 


7- 


10 


— 25. 


No. 


8. 


9 


+ 68. 


No. 


9. 


9 


+ 70. 



cent. lb. 


cent. lb. 


cent. lb. 


No. 1. 9 + 38. 


No. 4- 9 + 50. 


No. 7. 10 — 25. 


No. 2. 9 -f- 44. 


No. 5- 9 + 55- 


No. 8. 9 + 68. 


No. 3. 10 — 20. 


No. 6. 9 -f- 56. 


No. 9. 9 + 70. 


Ans. L. 186, /?. 4, d. 


I0l V 25 - 




-(- means more, — means less. 





THE ESSENTIAL FEATURES 



55 



from the marks placed on packages to designate excess and 

deficit with respect to listed weight. 

Van der Scheure, in his Arithmetica (1600), fol. Zi verso, 

defines the symbol + and — as signs of operation thus : 

-f- Plus Soubstraheert 1 
— Minus Addeert. 

He lapses, however, into using -r-, an old form of the minus 
sign, in the solution of his problems. 2 The use of these signs 
is to indicate operation, and their algebraic meaning, when em- 
ployed in equations, is seen in Tnierfeldern. 3 

The lack of these symbols made tabulation in sentence form 
impossible without the. use of words. Hence, the tabular facts 
of addition, subtraction, and also of multiplication were ex- 

1 This spelling for subtract is not an accident. The title of the chapter 
is Substraotio, fol. B 2 verso. It was quite common in the Dutch books of 
that time to spell .subtraction " substraction," a spelling not unheard of 
to-day and declared erroneous by lexicographers. 

2 Van der Scheure, Arithmetica (1600), fol. z x verso. 
"-]- Plus Soustraheert. 

" — Minus Addeert. 

"Soo 9. Eyers -j- 2. blancken soo veel weert zijn als 12. blancken -'- 21. 
Eyers/ Hoe veel Eyers coopt men dan om een blancke." 

If 9 eyers and 2 blancken are worth as much as 12 blancken minus 21 
eyers, how many eyers are worth as much as one blancke ? 

12 -f- 21 9 + 2 1 

2 21 

10 30 Facit 3 Eyers. 

3 Thierfelidern, Arithmetica (1587), page no. "Item/ 18 ft. weniger 
85 gr. rnachen gleich so vil als 25 ft. -=r- 232 gr. wie vil hat 1 ft. groschen? 
facit 21 gr." 

18 florins minus 85 groschens are equal to 25 florins minus 232 groschens, 
how many groschens are there in 1 florin ? Ans. 21 gr. 

In disen beyden Exempeln (he has given another example)/ addir das 
Minus/ und subtrahir das Plus/ wie hie : 

18/. -f- ^ gr. gleich 25/. -4- 232 gr. 
+ 85 



18/. gleich 25/. 


-*- 147 gr. 


%$fl. +147 gr. gleich 25/. 
18 




147 gr. gleich 7 /. 
7 /. 147 gr. 


1/. 



facit 21 gr. Page 110. 



5 6 SIXTEENTH CENTURY ARITHMETIC 






pressed in words or in' ruled tables according to a chosen 
device. 1 This condition of affairs in the formative period of 
arithmetic is responsible for the ruled tables still found in 
modern' arithmetics, 300 years after the necessity for them has 
disappeared. Some of them should be retained, doubtless, be- 
cause of their suggestiveness in showing number relations, but 
many of therm might be omitted to advantage. 

As in the case of addition:, there are many instances of be- 
ginning with a concrete problem. 2 Some writers who began 
addition with abstract problems began subtraction with the 
concrete ones. The following will illustrate: 3 "A goo 
man owed 800,347 livres, of which he has paid 409,- 409653 

653 livres : I wish to know how much he still owes." 

These problems are usually real situations, not con- 39 ° 
crete merely in the sense of being subtraction of denominate 
numbers. The problems given below from Champenois illus- 
trate this tendency and also the care taken in grading the steps 
in the process : 4 

1 See page 50, of this article. 

2 See page 48, of this article. 

3 Trenchant, L'Arithimetique (1578 ied.). "Vn homme doit 800347 1'fur 
quoy U en paye 409653 liures : fi ie veux fcauoir combien il doit de refte." 
Pol. B 8 recto. 

Dette 800347 
Paye 409653 



Refte 390694 

4 Ohaimpenois, "Les Institutions De L'Airothmetique " (1578). 
" Vn marohant a 58786 liures pefant de merchandif e, & en vendu 35040 
liures. On demande combien il a de refte." Page 17, or fol. C x recto. 

" Le Commis general des viures a 478759 pains, & en diftribue 27000 
pains. On demande eomlbien lil en de refte." Page 18, fol. C t verso. 

" Vn Architect a marohande f aire vne niuraiHe qui contiet 876 toifes, en 
a faict 374 toifes. On diemande combien il en a encor a faire." Page 19, 
fol. Cij recto. 

" Le Commis des viures du camp du Roy a 548 muids de ble, def quels il 
en a diftribue 273 muids. On demande cobien il en a encor' de refte." 
Page 20, fol. Cij verso. 
4 

£48 
273 

275 



THE ESSENTIAL FEATURES 57 

" A merchant had 58,786 livres weight of merchandise and 
sold 35,040 livres. It is required to find how much he had 
left." " 

" The commissary-general had 478,759 loaves of bread and 
distributed 27,000 loaves. It is required to find how much he 
had left." 

" An architect had bargained to make a wall which should 
contain 876 toises, of which he had made 374 toises. It is re- 
quired to know how much he had still to make." 

" The steward of a royal camp had 548 measures of grain, 
of which he had distributed 273 measures. It is required to 
find how much still remains." 

Multiplication 
Two classes of writers may be distinguished easily by com- 
paring their methods of treating multiplication. There were 
those who emphasized the formal processes themselves, and 
those who considered chiefly the applications of the processes. 
The former class of writers made much of tabular forms and 
devices, the latter made much of simple rules and commercial 
problems. Both gave the multiplication tables at the outset, 
which may be classified into three kinds : tabula per colonne, 1 

1 Rudolff, " Kunstliche rechnung mit der Ziffer und mit den zal pfen- 
aiige/ (1534 ed.), fol. Av verso. 



I 


1 




















2 


2 






2 


4 












3 


3 






3 


6 






3 


9 




4 


4 ' 






4 


8 






4 


12 




1 mal 5 : 


ist 5 


2 


mal 


5 ist 


10 




3 mal 5 ist 


15 


and so on to 9 


6 


6 






6 


12 






6 


18 


mal 9 ist 81. 


7 


7 






7 


14 






7 


21 




8 


8 






8 


16 






8 


24 




9 


9 






9 


18 






9 


27 




Borgi, Arithmetica (] 


[540 


ed.), 


fol. A 6 


verso. 








1 via 1 


sa 1 


2 via 


3 sa 


6 


3 


via 


4 sa 12 




8 via 9 sa 72 


2 via 2 


sa 4 


2 


via 


4 sa 


8 


3 


via 


5 sa 15 




8 via 10 sa 80 


3 via 3 


sa 9 


2 


via 


5 sa 


10 


3 


via 


6 sa 18 








9 via 10 sa 90 
















10 via 10 sa 100 


2 


via 


10 sa 20 


3 


via 10 sa 30 







In the same way he gave tables of 16s, 20s, 24s, 32s, 36s. 



53 



SIXTEENTH CENTURY ARITHMETIC 



or column tables; the tables ruled in squares, 1 or square tables; 
and the tables arranged in triangles, 



2 or triangular tables. 



Tartaglia gave the tables — 

o. fia o. fa o i. fia o. fa o 

o. fia i. fa o i. fia i. fa i 

o. fia 2. fa o I. fia 2. fa 2 



and so on to 
io. fia io. fa ioo 



o. fia. io. fa o i. fia io fa io 

These tables were set apart to be learned. Then followed the tables of 

us, I2s, 13s, 40s for reference, and the first set of tables with the 

middle numbers ten times as large; that is, from o. fia 0. fa to 10. fia 
100. fa 1000. These latter were next combined thus : 

11. fia 20. fa 220 20. fia 10. fa 200 

11. fia 30. fa 330 and so on to 20. fia 20. fa 400 



11. fia 100. fa 1 1 00 20. fia 100. fa 2000 

Finally he completed the tables from II. fia II. fa 121 to 20. fia 20. fa 
400. Thus, 

II. fia 11. fa 121 12. fia 12. fa 144 

11. fia 12. fa 132 12. fia 13. fa 156 and so on to 



11. fia 20. fa 220 12. fia 20. fa 240 20. fia 20. fa 400 

The tables of 12s, 20s, 24s, 25s, 32s, and 36s of this list he called " Per 
Venetia," because they were used in reckoning with Venetian money. 

1 Tonstall, De Arte Supputandi (1522), fol. G„ recto. 





;?! S> 4! Bl 6i 71 8 


Q ! iO| 


- 


3 3 


8 10 12 14 16 


is' So 


9 


ts\ 


12 16 18! 21 124 


27j 301 


-, 


3 12 


;•:■_; 20 24,2s: 32 


36 


40 


B 


10 15:20(25 SO'S5i40 


45 


CO 


c 


12 13; 84 'SO 35 42 48 


54 


60 


"J 


14 21 '28 135.42:49, 56 


63 


70 


3 


13|24=S2'40 43 53 i 64j72 


SO 


r& 


m\ 


SS 45 54 Rai 72 181 


90 


•10|5Qie0l7QIB0lfiQ 'ICO 



CirveJo, Tractatus Arithmetice practice (1513 ed.), fol. Avi recto. 



y 


8 


S 
8 


7 

7 


6 
1 6 


5 


4 [a 


S 11 


1 


9 


6 


4 3 


2 1 


2 


13 16 14 


12 10 


8 16 


4 


3 1 27 '24 21 


"28(15 


12 9 




4] 


S3 32 28 


24| 20 


16 




5 


45 40 35 30 25 




6 


54 48 42,36 






h 7 


63i 56.4a 






8 

,6 


72 


54 







* Topograph ical errors in the original. 



THE ESSENTIAL FEATURES 



59 



The column tables were used by the best commercial writers, 
and occasionally by the theoretic writers. The square arrange- 
ment, called the Pythagorean table, was used generally by 
authors of Latin School arithmetics. The triangular arrange- 
ment, constructed by some from left to right and by others 
from right to left, as shown in the notes, was the one in gen- 
eral favor. It will be noticed in the triangular table of p. 58, 
that the products 2 times 3 = 6, 3 times 4 = 12, and so on, 
appear, but that 3 times 2 = 6, 4 times 3 = 12, and so on, 
do not. Since any product, as 2 times 3, in one of these sets 
was deemed sufficient to represent itself and the corresponding 
product, as 3 times 2, in the other set, it is plain that writers 
recognized the commutative law of multiplication. The rows 
in the triangular table begin with square numbers. Gemma 
Frisius, 1 a famous Latin School writer, called attention to this, 
and Ramus 2 said that the pupil should first learn to multiply 
1 Gemma Frisius, Arithmeticae Practicae Methodus Facilis, fol. B verso. 



H IB j3j 41 6! 6 


, 7 L 8 


9 


J 


I 4 l 6! 8,10112 


141 16 


18 


2 


| 9 !12! 15! 18 |21 124 


27 


a 


ll6|20|24 


28 ! 32 | 38 i 4 


25 


30 


35140 ! 45 ; El 




38 41 


_s_ 






44 


bS 


63 


7 






6^ 


72 


3. 








81 



Qua- 
dra- 
ti- 
nu- 
me- 



2 Ramus, Arithmeticae Libri Duo (1586 ed.), fol. A ? verso, says that the 
pupil should first learn to multiply single numbers by themselves, as twice 
2 are 4, 3 times 3 are 9, 4 times 4 are 16, and so on, then the multiplica- 
tion of each single number with other single numbers as twice 3 are 6, 
twice 4 are 8, twice 5 are 10, and so on, but more attention should be given 
to larger numbers, as 9 eights are 72, 9 sevens, sixes, fives are 63, 54, 45, 

,8 sevens, sixes, fives are 56, 48, 40. 

" E notis autem multitudinis perdif cat primis fingulas per fe multiplicare : 
Bis 2 funt 4 Ter 3 funt 9 

Quater 4 funt 16 Quinquies quina funt 25 

Sexies 6 funt 36 Septies 7 funt 49 

Octies 8 funt 64 vies 9 funt 81 

Huiusmo'do multiplicatio quadratura dividitur, & numero factus hoc modo 
quadratus, factor autem latus quadrati. Fitqj ut numerus fecundum fuas 
unitates pofitus & additus tantundem faciat, quantum per fe multiplicatus, 
ut 2 & 2 faciunt 4: & bis 2 faciunt item 4. Sic 3 & 3 & 3 faciunt 9. 
& ter 3 faciunt item 9. Analogia etiam in talibus est continua, ut 1 ad 
factorem five latus, fie latus ad quadratum, ut in primo exemplo, ut 1 ad 



60 SIXTEENTH CENTURY ARITHMETIC 

the single numbers by themselves, as twice 2 are 4, 3 times 3 
are 9, and so on. This shows what the disciplinary teachers 
of that time regarded as important. 

The utilitarian writers, like Riese, Rudolff, and Kobel lim- 
ited the elementary products to 9 X 9 or 10 X 10, occasion- 
ally including the tables of twelves. It was more definitely 
stated that these facts should be learned * than in the case of 
the elementary sums. Tables given beyond 10 X 10, as the 
12s, 15s, 20s, 24s, usually related to the reduction of denom- 
inate numbers. 2 Thus, there were 12 denarii in 1 soldus, 20 

2, fie 2 ad 4. Turn fingularum notarum cum fingulis multiplicatione fciat 
quid efficiatur. 

Bis 3 funt 6: & ter 2 funt item 6. 

Bis 4 funt 8: & quater 2 tantundem. 



Octies 9 funt 72; & novies 8 tantundem." Fol. A ? verso. 
1 Riese, Rechnung auff der Linden und Fedem/ (1571 ed.). 
" Vnd du muft vor alien dingen das Ein mal ems wol wif fen/ und aus- 
wendig lernen/ wie hie." Fol. Avi verso. 
Kobel, Zwiey riechenbuchlin (1537 ed.). 

" Lern auswendig das Ein mal ein 
So wird dir alle Rechnung gmeyn." Fol. E verso. 
2 Cataneo, Le Pratiohe Delle Due Prime Matematiche (1547 ed.). 

DEL MULTIPLICARE LIRE, SOLDI ET DENARI. 

" Et fe ti fuf fe detto multiplica L 36. P 12. & dena. 7 per 

36.12.7 9. Segnato ohe harai le tue quantita come in margine, et tu 

9 multiplica 7. uia 9. che fa 63. dena. che per eifere ogni 12. 

denari un foldo i detti denari 63. faranno foldi 5. & dena. 3. 

329.13.3 di che fegnerai li. 3. denari & faluerai li 5. /3. dipoi multiplica 
9. uie 12. & aggiugneli il 5. faluato & fara /3 113. che per eifere 
ogmi foldii 20 una L i dette /3 113. fo«no L 5. & /3 13. onde feg- 
nerai li /3 13. & faluerai le L 5. Dipoi multiplica 6. uie 9. 

& aggiugneli il 5. faluato & fara 59. L dellequali fegnerai 9. 
faluerai 5. Poi multiplica 3. uie 9. & aggiugneli il 5. faluato 
& fara 32. ilquali fegna come da lato & iharai L 329. /8 13. et 
denari 3. per la detta multiplicatione. 

DEL MULTIPLICARE, MOGGIA STAIA ET. QVARTI. 

"Et dicendofi multiplica moggia 35. ftaia 11. & quarti 3. per 

15. Segnato che harai le quantita come dal lato e tu multiplica 

3. uie 15. che fa 45. quarti, perche ogni 4. quarti fanno uno 

35.11.3 ftaio, i detti quarti 45. feranno. ftaia 11. & un quarto, onde 

15 fegnerai un quarto & faluerai le 11. ftaia. Dipoi multiplica 

11. uie 15 & aggiugneli lo 11. faluato & fara ftaia 176. che per 



532. 8.1 effere ftaia 24. il moggio, le dette ftaia 176. fono moggia 7. 



THE ESSENTIAL FEATURES 6 1 

soldi in i lire, 24 staia in 1 moggia, and so on. Tartaglia 
(1556) calls these tables "Per Venetia," because they were 
based on the system of Venetian measures. The theoretic 
writers often filled in other tables, which were not of use in 
denominate numbers, in accordance with their policy of em- 
phasizing pure and formal arithmetic. 

In the formal process of multiplication more care was taken 
to grade the presentation than in addition and subtraction. 
Easy-graded steps appear in some of the earliest printed arith- 
metics. For example, Piero Borgi (1484) began with multi- 
pliers of one figure; he next gave problems in which the 
multiplier was a small number of two figures; then one in 
which it was a number of two figures ending in zero; then 
some with multipliers of three figures, and so on. It will be 
noticed that the multiplication by multipliers of two figures, 
as shown in examples x 2, 3 and 4 in note 1, was often accom- 
plished without partial products. This was done by referring 
to the corresponding tables. See notes, pp. 57, 58, under Borgi 
and Tartaglia. The following eight methods from Paciuolo 
( 1494) show with what mastery the leading scholars of arith- 
metic handled the Hindu algorism in the fifteenth century : 2 

& ftaia 8. di che fegnerai le ftaia 6. faluerai le moggia 7. 

" Dipoi mukiplica. 5. uie. 15. et aggiugneli il. 7. faluato & 
fara. 82. delquale fegnerai. 2. et faluerai. 8. Poi multiplier .3. 
uie .15. et a. quel che fa aggiugnelo .8. faluato et fara .53. 
quale fegna come in margine & harai moggia .532, ftaia .8. et 
un quarto per lo detto multiplicamento per uia del quale & del 
antedetto ti fera facile il multiplicamento de gli altri anoho 
che uariati pefi o mifure fuffero." Fol. Biiij verso. 
1 Piero Borgi, Arithmetica (1540 ed.), fol. B verso, B 2 recto and verso. 
1st 25 54 795 2d 345 3d 3456 4th 3456 

3 7 9 12 20 24 



75 378 7155 4140 69120 82944 

2 Paciuolo, Suma de Arithmetica Geometria Proportioni et Proportion- 
alita (1523 ed.). 

" Ora e da dire e mostrare in quati modi quefto aoto del multiplicare 
per la practica operatiua fe coftumi fare. Per laquel cofa dico che fimili 
acto de multiplicare f i coftuma fare principalmente in octo modi : di quali 
el primo e detto multiplicare per fchachieri in vinegia ouer per altro nome 
per bericuocolo in Firenza. El fecondo modo di multiplicare e detto caftel- 



62 SIXTEENTH CENTURY ARITHMETIC 

1. Per schachieri, as called 'in Venice, meaning tesselated. 
In Florence it was called bericuocolo. 

2. Castellucio. 

3. A Taveletta, or Per Colonna (by tables). 

4. Per crocetta (crosswise). 

5. Per quadrilatero' (in form' of a rectangle). 

6. Per gelosia, 1 or graticola (lattice-work). 

7. Per repiego (breaking up = factoring multiplier). 

8. A scapezza (distributing = separating multiplier into 
addends ) . 

1. CE Multiplieatio bricuocoli vel fchacberij. 

(Multiplication bricuocoli or schaoherii. Tesselated form.) 

Fol. 26 recto, or Dij recto. 
Multiplicandus 9^76 

Producendus 6 H $ 9 

Multiplicand j8 |8 \$\ 8 |4 | 

| 7|9|0|0|8| 
1 6 | 9 I 1 I 3 I 21 rchachieri 

1 5 1 9 I 2 I 5 1 6 I bericocolo 

Suma 67048164 p A. 

2. CE De .2°. modo multiplicandi dicto caftellucio. 

(The second method of mulitiplica^tion called castelluocio.) 

Fol. 27 recto, or Diij recto. 
9876 6 Per .7. 

6 
6789 1 Proua 



61101000 
Castelluccio. 5431200 The proof in each case is that of 

476230 casting out sevens. 

40734 






Suma. 67048164 .1. 

The chief feature consists in running the partial products to 
the right and filling the vacant places with zeros. 

lucio. El terzo e detto multiplicare per colona ouer a tauoletta. El quarto 
modo die lo multiplicare e detto per crocetta : e altramente per casfelle. 
El quinto modo e detto per quadrilatero. El fexto modo e detto per 
gelofia: ouer gratkxla. El feptimo modo e detto per repiegO'. Loctauo 
modo e detto multiplicare a f capezzo." Fol. Diij recto, or 26 recto. 

1 So called because Italian ladies were protected from public view by 
lattice-work over their windows. 



THE ESSENTIAL FEATURES 



63 



.3. CE De tertio modo multiplicandi ditto 'colona, 

(The third method! of multiplication called oolonna. By tables.) 

Fol. 27 verso, or Diij verso. 



4685 
13 



Per .7. 
2 
6 



60905 
Proua 



No partial products were needed in this method, since a 
table of 13s was presupposed. 

4. (I De quarto mo multiplicandi ditto crocetta five cafella. 

(The fourth method of multiplication called crocetta or casella. 
Crosswise multiplication.) Fol. 27 verso, or Diiij verso. 



13 6 9 



2 7 9 3 6 



(I De quinto modo multiplicandi dicto quadrilatero. 

(The fifth method of multiplication 'Called quadrilatero. 
form of a rectangle.) Fol. 28 recto, or Diiij recto. 



In the 



54 8 8 
5 4 3 2 



I 





8 


6 4 


1 


6 


2 


96 


2 


I 


7 


2|8 


2 


7 


1 


6|0 



2 9 6 6 



29506624 Suma 



6. C De fexto modo multiplicandi dicto gelofia: siue graticola. 

(The sixth method of multiplication called gelosia, or graticola. 
Lattice- work.) Fol. 28 verso, or Diiiij verso. 



\3 


5\ 


\9 
4 \ 


\ 2 

7\ 


\4 
6\ 


\6 

e \ 


\1 


\2 


9 


V 


4 



7. (L De feptimo modo multiplicandi ditto repiego. 

(The seventh method of 'multiplication called repiego. That is, 
multiplication foy the factors of the multiplier separately.) 



64 SIXTEENTH CENTURY ARITHMETIC 

6 x may be broken up into the factors 2 and 3. Since 2 
times 3 make 6, the factors of 6 are 2 and 3. The factors of 
10 are 2 and 5, since 2 times 5 make 10. It often happens that 
a number may be factored in different ways; for example, 12 
has different groups of factors; it has the factors 2 and 6, 
since 2 times 6 are 12, also the factors 3 and 4, since 3 times 4 
make 12. And thus 24 has several groups of factors, as 12 
and 2, 3 and 8, and 4 and 6, which multiplied together 
make 24. 

8. d De octauo modo multiplicandi dicto a fchapezzo. 

(The eighth method of multiplication called a fchapezzo, distrib- 
uting.) 

When 2 42 is to be multiplied by 24, one of these numbers 
(it makes no difference which) may be resolved into several 
parts. As 24 may be resolved into four parts which added 
together make the whole number, as 4, 6, 5, 9, then commence 
with any one of these and multiply by 42. For instance, take 

4, and 4 times 42 makes 168. Place this aside. Then 6 times 

1 Paciuolo, Suma de Arithmetica Geometria Proportioni et Proportion- 
alita (1523 ed.). 

" Si commo de .6. diremmo esfer el. 2. e. 3. Perche .2. via .3. fa .6. fi 
che el repiego de .6. e .2. e .3. El repiego de .10. e .2. e .5. perche .2. via 

5. fa 10. E acade molte volte vn numero hauer asfai repieghi varij e 
diuerfi : f i corno .12. ane piu repieghi : poche hane el repiego de .2. e .6. 
che .2. via .6. fa .12. Ane el repiego de .3. e .4. che .3. via .4. fa 12. E cosi 
.24. a piu repieghi/ cioe .2. e .12. e .3. e .8. e .4. e .6. ehe luno elaltro mul- 
tiplicato fa .24. : cioe per .6. e di .6. via .116. fa 696. 

Fol. 28 verso, 29 recto, or Diiij verso and Dv recto. 

2 Paciuolo, Suma de Arithmetica Geometria Proportioni et Proportion- 
alita (1523 ed.). 

" Si commo hauendo a multiplicare .42. via .24. dico che ne refolua vno 
de quefti numeri qual voli (che non fa cafo) in piu parti acio te fia piu 
co modo el multiplicare. Or fia che tu refolua .24. in quatro parti che 
f ieno luna .4. laltra .6. laltra .5. laltra .9. Dico che comenzi daqualuoli : e 
multiplicata via .42. Or fatte dal. 4. e di .4. via 42. fa .168. Qual metti da 
canto. E poi .6. via .42. fa .252. e poi falua lotto .168. de ritto luno e 
laitro: cioe numero fotto numero edicine lotto dicie^. E poi di .5. via .42. 
fa .210. e falua lotto le altre. E poi dirai .9. via .42. fa 378. Qual simil- 
mete falua fotto laltre e recogli mo tutte quefte .4. multiplicationi infieme 
oioe .168. 252. 210. e 378. faron© .1008. e itanto dirai che facia .24. via .42." 
Fol. 29 recto or Dv recto. 



THE ESSENTIAL FEATURES 65 

42 makes 252. Save this with the 168, etc. Then 5 times 42 
makes 210. Place this with the others. Finally, 9 times 42 
makes 378. Put this also under the others in the same man- 
ner, and then collect the four multiplications together, as 168, 
252, 210 and 378 making 1008, which is the product of 24 
and 42. 

Tartaglia (1556) and Unicorn 1 (1598) each gave seven of 
the above methods, which shows not only Paeiuolo's influence 
upon his countrymen, but also the tenacity with which theo- 
retic writers held to disciplinary arithmetic. 

Besides the above general processes, there are several par- 
ticular ones of interest, such as complementary multiplication, 
arrangement of factors, and order of multiplying. The fol- 
lowing is an example of complementary multiplication for find- 
ing the product of two digits, as given by Cirvelo (1513). 2 
This was his plan of multiplying 6 by 8 : 

2 (the complement of 8) X 6 = 12 6 

8 

60 (= 10 X 6) — 12 = 48 48 

A variation of this process was as follows : To multiply 7 by 8 : 

7.3 10 — 7 — 3 Multiply the complements 3 and 2 (= 6). 

8.2 10 — 8 = 2 Add the numbers 8 and 7, using only units' 

figure in the result. 

56 

Such multiplication was not commonly used. Riese justified 
its use because it was an available method for those who have 
not learned the tables. 3 

1 Unicorn, De L'Arithmetica Universale (1598 ed.), often gives credit to 
Paciuolo. 

2 Cirvelo, Tractatus Arithmetice practice (1513 ed.)- "Verbi gratia: 
oeties fex faciunt .48 nam octo dif tat a decez per duas vnitates : ergo fub- 
trahitur .6. de fexaginta q e fua dena bis et remanebunt .48. Fol. a v recto. 

According to Cantor, complementary multiplication was used by the 
Romans. It is possible that it antedates the invention of the abacus. 
Weissenborn, Gerbert, 171 -2. 

3 Riese, Rechnung auff der Linien und Federn/ (1571 ed.). " Leret 



66 



SIXTEENTH CENTURY ARITHMETIC 



Tonstall x said that, if the numbers are unequal, the larger 
should be placed above as the multiplicand. In ,^ , 2 ^ 

the case of 185 times 13 bu., this arrangement 185 185 

would lead to a concrete multiplier, a form 13 bu. 13 
guarded against in modern teaching. But this 
was avoided by omitting all denominations from the numbers 
when used in calculation, as in example 2 above. The proper 
denomination was affixed to the result when obtained. Sev- 
eral examples of multiplication exist in which the higher 
orders of the multiplier are used first, the 45 6 7 3 

method da Fiorentini of Tartaglia being 
an example. The Castelluccio method of 
Paciuolo differs from these only in having 
the multiplier written above the multipli- 
cand. There seems to have been no use at 
that time for beginning with the highest 
order of the multiplier. But, after the decimal fraction was 
introduced, this plan found a useful application in making ap- 
proximations. For example, in the work in the 1.26 
margin the part at the right of the vertical line need 
not be calculated if the result is needed to tenths only. 

Tests were prominent in multiplication, as in other 
operations, the proof by nines and the proof by 
seven being preferred. The example from Tartaglia at the 
top of p. 67 illustrates the proof by casting out sevens. Divi- 



4326 





18268000 




1 3701 00 




91340 




27402 




19756842 











2.35 



2.5 

•3 
.0 



2.9 



2 

08 

630 



610 



viel machen/ Muft audi forne anheben/ Vnd fur alien dingen das Ein unal 
eins/ auswendig tern-en/ wie vorhin angezeiget/ oder mache es nach fol- 
genden zweyen Regulen." Fol. Bv recto and verso. 



8.2. 
9.1. 


I 7-3- 
J 8.2. 




6.4. 
8.2. 




6.4. 
7-3. 


7.2. 


1 5-6. 




4.8. 




4.2. 



The rule here suggested was known- as the sluggard's rule. 

Noviomagus (1544), Gemma Frisiius (1540), Baker (1580), gave the 
second method. 

1 Toiiistall, De Ante Supputandi (1522 ed,). "Et fi numeri lint in- 
aequales: maior Temper supra pro multiplicado ponatur: minor infra pro 
multiplicanti." Fol. G verso. 



THE ESSENTIAL FEATURES 67 

sion was occasionally used to prove multiplication, 504 

although the explanation of division constituted a 24 3 

later chapter. 1 12096 _o 

The short methods, although common, were con- 
fined to three classes : (a) the use of factors in the multiplier, 
(b) multiplication by multipliers ending in zero, and (c) 
cross-multiplication. 

(a) Multiplication by using factors of the multi- 87 
plier. 3 

This plan has been illustrated already in Paciuolo's 261 

method, called repiegO', number 7, page 63. The fol- 3 

lowing occurs in Trenchant : 2 To multiply 87 by 9. 783 

(b) Multiplication by numbers ending in zeros. 

Piero Borgi, in multiplying 3456 by 20, gave the following 
explanation: 3 6 X 20 = 120, then 5 X 20 = 100, 
100 -f- 12 = 1 12, of which the 2 belongs to tens' place; ' 20 



4 X 20 = 80, 80 + 11 = 91, of which the 1 belongs — 

to hundreds' place; 3 X 20 = 60, 60 + 9 = °9> the 
whole result is 69120. In his second method of multiplying 
by 20 he first multiplied by 2 and then by 10. Philip Calandri 
(1491) took up multiplication by 100 as a special case, giv- 
ing problems about 100 oranges, 100 chickens, 100 calves, 
and various things. Tonstall directed placing at the right of 
the multiplicand as many zeros as there are in the multiplier. 

1 Riese, Reohinung auff dier Linien mid Federn/ (1571 ed.), fol. Bvii recto. 

2 Trenchant, L'Arithmetique (1578 ed.), fol. C i recto. 

3 Borgi, Qui oomeza la nobel opera de arithmeticiha (1540 ed.). 

" E fe hauefti ia anolitiplicar .3456. per .20. prima metiterai le due figure in 
forma, poi cominciando 'dalle vnita dirai .6. via .20. fa .120. che 
fono apunto .12. defene fenza foprauanzo de vnita, & pero in 3456 

luogo delle vnita metterai .0. e dirai nulla e tien .12. defene, poi 20 

alle defene .5. via .20. fa .100. e .12. che tenefti fa .112. che fono 

.II. centenara e .2. defene e metterai le defene a fuo luogo. e dirai 69120 

.2. e tien .11. cetenara poi alii cetenara dirai .4. via .20. fa .80. e II. 
che tenefti fa .91. che fono 9. miara e vn centenaro, e metterai il centenar 
a fuo' luogo, e dirai .1. e tie .9. miara, poi alii miara dirai .3. via 20. fa .60. 
e .9 ohe tenefti -fa .69. ilql metterai a fuo luogo appreffo il. 1. fara 69120 
adoque moltiplicato .3456. p. .20. fa 69120." Fol. B 2 recto. 



68 SIXTEENTH CENTURY ARITHMETIC 

This method was also used by Rudolff and Car- 36 

dan. 1 Gemma .Frisius, 2 in multiplying two num- 7? 

bers, as 3600 by 7200, rejected the zeros, multi- 72 

plied as usual, and then annexed the zeros to the 252 



result. This method was also used by Baker. 3 2592100001 

(c) Cross-multiplication, known as per cro- 

cetta, or per crosetta. 1x1x1 

The following example is taken from Tar- 456 3 

taglia. 4 This method was also used by Paci- * 48200 — 3 

3 3 
uolo, Unicorn, Borgi, and several others who 

followed the Italian School. 5 

Cardan gave the following methods for aiding the memory 

in multiplication: 

1. To multiply 27 by 33: 

2 7 + 33 = ^o 60 -T- 2 = 30 30 2 = 900 

30 — 27 = 3. 3 2 = 9 900 — 9 = 891=27X33- 

2. To multiply 27 by 63 : 
27X6=162 27X3 = 81 
1620 + 81 = 1701 = 27 X 63. 

3. To multiply 37 by 49: 

40X50 = 2000 40 — 37 = 3 50 — 49=1 
2000 + 3 = 2003 1 X 40 = 40 
3 X 50=150 + 40=190 
2003 — 190 = *28i3 = 37 X 49- 

4. To multiply multiples of 10: 
30 X 70 = 2 1 hundreds 

700 X 800 = 56 ten thousands = 560000 
17 X 70= 119 tens= 1 190. 

Many writers of commercial arithmetic, and even some 
Latin School writers, as Gemma Frisius, proposed a concrete 

1 Cardan, Practica Arithmetice (i539 ed.), fol. Bvi verso. 

2 Gemma Frisius, Arithmeticae Practicae Methodus (1581 ed.), fol. B 3 
recto. 

3 Baker, The Well Spring of Sciences (1580 ed.), fol. Dvi verso. 

4 Tartaglia, Tvtte L'Opere D'Arithmetica (1592 ed.), fol. E 6 recto. 

5 Cross multiplication is one of the six methods given by Bhaskara in. 
the Lilivati. 

* Error in original. 



THE ESSENTIAL FEATURES 69 

example in multiplication before explaining the process. 
Calandri 1 gave as his first example : " Multiplica 9 vie 7389 
# 11 /? 8 d. (Multiply 9 by 7389 y 11 p 8 d.)" Gemma 
Frisius, for his first example with a multiplier of two figures, 
gave this example : " I wish to reduce 267 days to hours.'' 2 
The following are from Champenois: "A squadron has 312 
men in rank and 232 in file; how many men are there in the 
squadron?" 

"A wall is 1212 toises in length and 4 toises high; how 
many toises are there in the wall ?" 3 

Division 

The methods of division used at that time have a peculiar 
interest. Most of the methods of adding, subtracting, and 
multiplying that were in general use in the sixteenth century 
are used to some extent at the present time, but the method 
of division most commonly used then is entirely obsolete now. 
This was known as the scratch, or galley method. 4 

A simple example from Baker will give the principles of 
the method : 5 
To divide 860 by 4. The devidend. 860 



Dividend 



Deuisor. 



Divisor £ (2 quotient 

£ = (4X2) subtracting 8 from 8 leaves nothing to be placed 
above. 

1 Calandri, Arithmetica (1491 ed.), fol. 18 recto, or Cvi recto. 

2 Gemma Frisius, Arithmeticae Practicae Methodus Facilis (1581 ed.). 
" Exampli gratia, 267 dies valo redigere ad horas." 

3 Champenois, Les Instkvtions De L'Arithmetique (1578 ed.). 

"Vn efcadron contient en front 312 hommes, & en flanc 232. Lon de- 
mande combien il y a d'hommes en 1'efcadron." Page 27, or fol. Cvi recto. 

"Vne muraille contient en longueur 1212 toifes, & en hauteur 4 toifes. 
On demande combien de toifes contiet la muraille." Page 29, or Cvii recto. 

4 The galley, or scratch, method of division is doubtless an inheritance, 
having its origin in the sand-table calculation of the Hindus. Treutlein, 
Abhandlungen, 1 : 55. 

Maximus Planudes (c. 1330) also explains its origin in this way. Jour- 
nal Asiatique, Series 6, vol. 1, p. 240. 

5 Baker, The Well Spring of Sciences (1580 ed.), fol. Dviii recto and 
verso, Ei recto. 



;o 



SIXTEENTH CENTURY ARITHMETIC 



In the next step the divisor is moved one place to the right. 
2 subtracting 4 (= 4X1) from 6) 

mm 

4 4 is -contained in 6 once, so 1 is written in the quotient 

2 The 2 placed above is the remainder after having subtracted 

£>^0(215 4 from 6. This 2 in tens' place with the o still left in units' 

4 place forms 20. 20 -divided by 4 leaves 5, the last figure of 

20 the quotient. 

The following is an example of the scratch method from 
Tartaglia showing a remarkable form of galley : x 



8 
o 9 
i 6 
5 5 

O 

1 oi> 
± S S 

9 9 
9 



<) 
o 

7 6 

4 

6 6 

s s 

9 9 9 

9 9 9 



I ] ■ 



o 9 

O J- 
S 7 <? 

8000000 <r 9 p<4' 8 

0000000^6 (t 
0000000$$$$0 
000000009 9' 9P 
90000000O999 



9 9 



o S 6 

$ S 7 7 
0000000*9994: 
0000d00S<56666 

0O00000S5SSSSl8S 

0000000099999 I 

000000000^99 



Deltcrzo modo depart ire dctto a danda. 



4' nrrrromododipartircda noftriantichi pratici t detro a danda 4 quale pur generate, fi come \1 
p.irrn c per kuello,ouer galca.cioe che per tal modo fi puo parrire per ogni numero , ma in que-- 
(to non fi drpenna mai alcuna figura nel opcrare ,come fi fa nel partir per batelio , oner galea. ,& 
accio inrgjio !o apprcndi,poniamo ciic ru voglia parti're quel medefimo 91254?. per 1 987. che 



The downward method of the present day appeared 
among those used in the earliest printed arithmetics. 
This example is from Tartaglia : 2 

I auenimento (Quotient) 
partitore 1987 | 912345 | 459 
(Divisor) 



9123 
7948 
H754 
9935 



18195 
17883 



(Remainder) auanzo 312 

1 Tartaglia, La Prima Parte del general Trattato (1502 ed.), fol. Gv 
recto. 

2 Tartaglia, Tvtte L'Opere D'Arithmetica (1592 ed.), fol. G g verso. 



THE ESSENTIAL FEATURES 



71 



f r*?> — 8 ] 



O0C44-II 



7? 4. 

4^8 



7^ 



8? 



4-f 





ST 


e tOam | g- 


Co -parti n>i\ 


i — £0 


n>i — 


1/* 


\}>i/-fr 


+ao 


Uknnc 1 1 ti 


nichne t?o- 




•parti Co p 


| 'Parti > e -? 


Co — * 


? — \ 


48o 


LL *f/* 


uknnc 1 Co 


Uicimc f? 



1 1. 



Paeiuolo also grave it as one of his methods, called division 
" a danda." 

This illustra- 1 - — —- = ^ - ^ . 23 

tion is from Cal- *P3rn 

andri, 1 in whose 
book appears so Uictmc 

far as known, the 
first downward 
division ever 
printed, although 
it is found occa- 
sionally in manu- 
scripts of the fif- 
teenth century. 

The lists of 
writers given be- 
low show rela- 
tively the extent 
to which the gal- 
ley method 2 and 
the downward 
method 3 were 
used. Nearly all 
who used the 
downward meth- 
od also used the galley method, while many treated the 
galley method who did not explain the downward form. 

Besides these general processes there were several other 
forms. The method a tavoletta, variously called per colona, 
di testa, per discorso, and per toletta, was used by Paciuolo 

1 Calan-dri, Arithmetica (1491 ed.), fol. 33 recto. 

2 Among those who used the galley methods were: Borgi (1484) ; Wid- 
man (1489) ; Cirvelo (1513) ; Tonstall (1522) ; Paciuolo (1494) ; Rudolf! 
(1526); Kobel (1531) ; Cardan (1539); Noviomagus (1539); Tartaglia 
( J 556) ; Gemma Frisius (1540) ; Riese (1522) ; Ramus (1567) ; Trenchant 
( J 57i) ; Champenois (1578) ; Baker (1580) ; Raets (1580) ; Unicorn (1598) ; 
Van der Scheure (1600). 

3 The downward method was used by Calandri (1491) ; Paciuolo (1523 
ed.) ; Tartaglia (1556) ; Trenchant -(1571) ; Unicorn (1598). 



72 



SIXTEENTH CENTURY ARITHMETIC 



(1494), Tartaglia (1556), and Unicorn (1598). This is 
short division where the result of each part Can be taken from 
the table. Tartaglia gave as examples : 

Divisor 2) 1 7953 Divisor 12) 2 7630 

Quotient 3976 Remainder 1. Quotient 635 Remainder 10. 

The method a repiego (per repiego) was used by Paciuolo s 
(1494), Tartaglia, and Unicorn. In this method the divisor 
was separated into factors, as in the example : 4 To divide 
5867 by 48. 5867 -7- 6 = 733 with a remainder 3. 733 -r- 
6 == 122 with a remainder 1. 

Wencelaus gave a form called by him Italian division, of 
which the following is an example : 5 

To divide 11 664 by 48. He first divided the divisor, 48, 

tv • into halves, fourths, eighths and six- ~ .. 

Divisor ' ' & Quotient 

8 T teenths, the second group representing 

24 05 lj h h h rV- (The first zero evidently 00625 

12 02 5 shows the lack of units, and the 05 

~ °^ s figures beginning at its right repre- 
sent respectively tenths, hundredths, 
thousandths, and ten thousandths. It is a decimal system 
without the use of the decimal point, a device which 
did not appear until about 1600.) Beginning at the 
left of the dividend, 11,664, 11 is the first number 
that contains any one of the parts of the divisor as tabu- 
lated. The largest part which it contains is 6. The fraction 
which corresponds to it, 0125, is entered as part of the quo- 
tient, as tabulated at the right. 6 is then subtracted from 1 1 
and the remainder, 5, is treated similarly. Since 5 contains 

1 Tartaglia, Tvtte L'Opere D'Arithmetica (1592 ed.), fol. F 4 recto, 
a partir 2.//79S3 

ne vien — 3976 — e auanza 1 

2 Tartaglia, Tvtte L'Opere D'Arithmetica (1592 ed.), fol. F ? verso. 
a partir per 12// 7630 

ne vien 635 auanza 10 

s Paciuolo gave four methods of division : first, a Regola, or a tavoletta 
(iby table) ; second, per repiego (in parts) ; third, a danda (downward) ; 
fourth, a Galea or per galea (galley method). 

4 Tartaglia (1592 ed.), fol. G 6 verso. 

5 Wencelaus, T'Fondament Van Arithmetica (1599 ed.). 



THE ESSENTIAL FEATURES 



73 



2 

ffl 

nm 



012 5 

0062 

05 

05 





243 



3, the next part of the quotient is the corresponding fraction, 
tV» ot 00625. 3 from 5 leaves 2, which is not in the list of 
parts of the divisor, hence the next figure of the 
dividend, 6, is annexed, making 26. This contains 
24, so the corresponding 05 is written in the quo- 
tient, one place farther to the right, and so on. 
Whenever a new order of the dividend is used the 
partial quotient is set one place farther to the right. 
He gives as the complete form the example at the 
right. 

In the matter of detailed processes, Champenois was to 
French arithmeticians what Tonstall was to English writers, 
though less verbose. In his treatment of division Champe- 
nois gave twelve cases : * 
1. To divide a digit by a digit. 2 (Exact division.) 

1 Champenois, Les Institutions De L'Arithmetique (1578 ed.), Diij rector, 
or page 37 et seq. 

2 The division tables were not so common as the tables of multiplication. 
The inverse relation of division to multiplication was generally recognized, 
on which account one set of tables sufficed. A few writers who aimed at 
completeness gave tables of division. 

E. g., Tartaglia, La Prima Parte (1556), fol. Eiiij recto. 
i in o intra o e auanza 

(1 is contained in o, times with remainder o.) 
1 in 1 intra 1 e auanza o 
1 in 2 intra 2 e auanza o 



1 in 9 intra 9 e auanza o 

2 in o intra o e auanza o 
2 in 1 intra o e auanza 1 



9 in intra o e auanza o 
9 in 3 intra e auanza 3 



2 in 19 intra 9 e auanza 1 9 in 89 intra 9 e auanza 8 

Tonstall, De Arte Supputandi (1522 ed.), used this inverted form of the 
Pythagorean table, fol. Y 2 recto. 



100 1 90 1 80 1 70 1 60 1 so 1 40 1 30 1 20 1 


10 


90 | 81 | 72 1 63 1 54 1 45 1 36 1 27 1 18 I 


9 


80 1 72 1 64 1 56 1 48 1 40 1 32 1 24 1 16 | 


8 


70 1 63 1 56 1 49 1 42 1 35 1 28 I 21 | 14 | 
60 I 54 1 48 1 42 1 36 1 30 1 24 1 18 | 12 I 


7 
6 


50 1 45 L 4 ^-'_35 1 30 Ij25 1 20 1 15 | 10 1 
40 1 36 1 32 1 28 | 24 1 20 | t6| 12 J 8 | 


5 
4 


30 | 27 | 24 1 21 1 18 1 15 1 12 | 9 | 61 
20 1 18 1 16 1 14 1 12 1 10 | 8 1 6| 4 1 


3 
2 


10 1 9 I 8| 7I 61 5I 4l 3l 2| 


1 



74 SIXTEENTH CENTURY ARITHMETIC 

2. To divide a digit by a digit with remainder. 

3. To divide an article (a number ending in o) by a digit. 

4. To divide a number whose first figure is smaller than 
the divisor. 

5. To divide an article by an article. 

6. To divide a composite number (a number formed by com- 
bining an article and a digit) by an article. 

7. To divide a composite number by a composite number. 

8. To divide a number when the number left after any sub- 
traction is too small to be divided by the divisor, as 13 1328 

-^ 43 2 = 3°4- 

9. Inexact division. 

10. When the remainder is greater than the divisor it proves 
that the division is incorrectly performed. 

11. When the amount to be divided is less than the divisor, 
then fractions result. 

12. To divide a number by 2. 

Such development was not characteristic of the writers of 
that period. It was customary to begin with a dividend of 
several figures, but the methods of division were so radically 
different from our present ones that it is not safe to say that 
long division in the modern sense generally preceded short 
division. It is possible, however, to find indisputable cases of 
this plan. 1 

The proofs for division were casting out nines, casting out 
sevens, and the inverse operation. As has already been stated 
in this article, the prevalence of proofs was due to the influ- 
ence of abacus reckoning, and not so much to a sense of the 
need for verification. Kobel in Germany and Baker in Eng- 
land seemed to realize the uselessness of appending several 
proofs to each operation, for they gave no proofs until all the 
operations with integers had been presented. Kobel then 
gave the proof of nines for all operations, and Baker gave 
the inverse operations. It would be a decided improvement 
on present teaching to introduce both the proofs by nines and 
by the inverse operations where practicable. 

1 Trenchant, L'Arithmetique (1578 ed.), fol. D 3 recto et seq. 



THE ESSENTIAL FEATURES 75 

A few short methods were used in division, the most com- 
mon being those for dividing a number by 10, 100, and 1000. 
This example from Champenois will serve to illustrate : 1 
" 54736 livres are to be divided among 10 men/' The quo- 
tient was formed by removing the last figure of the dividend 
and making it the remainder. Baker also explained in the 
same way the division by 100, 1000, and 10,000. Tagliente, 
after his explanation of division by 10, also explained divi- 
sion by 100 and iooo. 2 "And if you wish to divide 3497 by 
100, do the same as above, taking off as many figures for a 
remainder as there are zeros in the divisor, as seen in the 
following division: 34I97." Similarly 749,745 by 1000, 
749 1 745. A list of practical problems was given by Calandri 
in which the cost of 100 things was known and the cost of 
one required. 3 

A more general case is the division of numbers by multi- 
ples of 10. For example, 4 to divide 5,732 by 573(2 
20, cut off the last figure of the dividend; 2 §6 with 
divide the part at the left by 2. Change this remainder 12 
remainder 1 to 10 and add the 2 cut off. Then 5732 divided 
by 20 gives 286 and the remainder, 12. 

Finaeus 5 wrote down the multiples of the divisor before 
performing the division. 

1 Champenois, " Les Institutions De L'Arithmetique (1578 ed.). 
"Come f'il falloit diuifer 54736 liures a 10 hommes, fault trancher le 

dernier nombre de la fomme a diuifer 6. Le refte 5473. donnera le Quo- 
tient. Parquoy 54736. a partir a 10. hommes, c'eft a chacun 5473. liures, & 
6. liures qui reftent a partir a 10 hommes." 5473(6 
Fol. Eiij verso. 1 (o 

2 Tagliente, Libro Dabaco (1515 ed.). "Et fe volefti partire 3497 per 
100 farai nel modo ditto difopra taglia tante figuare quanti .0. li a el tuo 
parti doc come tu vedi qui fotto e fata partito. 34.I97." Similarly 749745 
per 1000. 749 j 745. Note the bar used as a decimal point. 

3 Calandri, Arithmetica (1491 ed.). "Cento melarance choftorono 53 P 
4 d. che uiene luna." 100 oranges cost 53 /3 4 d., what is the cost of one? 
"Cento pollaftre (chickens) coftorono 26 ( y 10 ft che uiene luna." " Cento 
capponi (capons) coftorono 97 & 1 /3 8 d. che uiene luno." " Ceto uitelle 
(calves) coftorono 2354 y 10 /3 che uiene luna." Fol. 29 recto and verso, 
fol. (e) recto and verso. 

4 Tartaglia, Tvtte L'Opere D'Arithmetica (1592 ed.), fol. F 7 recto. 

5 Finaeus, De Arithmetica Practica (1555 ed.), fol. 9 recto. 



76 SIXTEENTH CENTURY ARITHMETIC 

Division, like the other processes, was occasionally intro- 
duced by concrete examples. Gemma Frisius follows his 
definitions of dividend, divisor, and the remainder with : * 
" 433 D 5^ aurei are to be divided among 72 men, what will 
each receive?" and Kobel's first example is: 2 " There are 5 
companions who must share equally 40 guldens. They would 
like to know what part each should have." 

Doubling and Halving 
A striking example of extreme subdivision and classification 
is the appearance of Duplatio (doubling) and Mediatio (halv- 
ing) in the list of Species. To the student of the present 
there seems to be no reason why doubling and halving should 
have been treated independent of multiplication and division. 
That the reason was not apparent to scholars of that day 'is 
shown by this statement from Gemma, Frisius : 3 " Some are 
wont to regard Duplatio and Mediatio as operations separate 
from multiplication and division. I do not understand what 
influences those 'stupid ones, since to double is to multiply by 
2 and to halve is to divide by 2. If these operations are dis- 
tinct, an indefinite number of operations will arise for con- 
sideration, as triplatio, quadruplatio, and so on." In discuss- 
ing the contents of the Bamberg Arithmetic (1483), Cantor 
says : 4 " The treatment of doubling and halving as partic- 
ular species or wholly unknown as such is the infallible sign 
of whether the writer belongs to the school of Jordanus or to 
that of Leonardo. The Bamberg Arithmetic treats of these 

1 Gemma Frisius, Arithmeticae Practicae Methodus Facilis (1581 ed.). 
" Ut fi diuidendi f int 433656 aurei 72 hominibus." Fol. B 4 recto. 

2 Kobel, Zwey rechenbuchlin (1537 ed.). " Es fein funff Gefellen/ die 
haben zu theylen 40. gulden/ und wolten gern wiffen wie vil iedem zu 
seinem theyl werden solt." Fol. F verso. 

3 Gemma Frisius, Arithmeticae Practicae Methodus Facilis (1581 ed.). 
" Solent non nulli Duplationem & Mediationem a fignare species diftinctas 
a multiplication e & divifione. Quid vero monuerit stupidos illos nefio, 
cum & finitio & operatio eadem fit, Duplare enim, eft per duo multipli- 
care. Mediare vero, per duo partiri. Quod Ti hae operationes lint dif- 
tinctoi, iniinitae jam nobis exorientur fpeoies, triplatio, quadruplatio &c. 
Sed fatis de illis." Fol. B g verso. 

4 Cantor, M., Gesohichte der Mathematik (3d ed., 1900), Bd. II, page 227. 



THE ESSENTIAL FEATURES 



77 



operations as special cases under the Species and knows noth- 
ing of them as separate species; thus it is an emanation of 
Italian teaching spreading to southern Germany." That these 
operations were accorded a high degree of independence is 
shown by the fact that Widman, Kobel, and Riese placed 
them before multiplication and division. The following may 
be taken as typical examples under these processes : 

41232 1 11 V- 15241578570190521J 3 

82464 43672136(21836068 Divisor 2 Quo. 7620789285095260 

22222222 

The first is an example in duplatio from Riese and the second 
an example in mediatio from Gemma Frisius. Gemma Fris- 
ius, however, treats mediatio simply as a case under division. 

DENOMINATE NUMBERS 

Having now discussed the four operations with integers, 
the order of the remaining subjects must be taken arbitrarily, 
since there was no uniformity in this matter in sixteenth cen- 
tury works. By some writers fractions and denominate num- 
bers i were both treated under operations with integers, and 

1 Riese, Rechnung auff der Linien und Federn/ (1571 ed.), fol. Biiij 
recto. 

2 Gemma Frisius, Arithmeticae Practicae Methodus Facilis (1581 ed.), 
fol. B Q verso. 

3 Widman, Behend und hupsch Redinung (1508 ed.), fol. Ci verso. 

4 Among those who treated the operations with integers separately were : 
Riese (1522) ; Tartaglia (1556) ; Borgi (1484) ; Widman (1489) ; Ton- 
stall (1522) ; Champenois (1578) ; Rudolf? (1526) ; Gemma Frisius (1540) ; 
Noviomagus (1539). 

Among those who treated operations with denominate numbers under the 
respective operations with integers were: Raets (1580); Unicorn (1598); 
Calandri (1491) ; Baker (1562) ; Van der Scheure (1600) ; Cataneo (1546) ; 
Cardan (1539). 

Cardan treated under each operation all phases: e. g., under addi- 
tion he treated addition of integers, denominate numbers, fractions, surds, 
powers, roots, similarly under the other operations. 

The expression "denominate numbers" was used by Cardan to denote 
powers and roots of numbers. Thus, 

cosa. census. cubus. census census. Relatum primum. 
248 16 32 

or the first, second, third, fourth, fifth, powers of 2 are given under the 
last section of Chapter I on Arithmetic, fol. Avi recto. 



yS SIXTEENTH CENTURY ARITHMETIC 

seldom were fractions and denominate numbers wholly separ- 
ated from each other. 

A thorough treatment of denominate numbers was import- 
ant, not only because of their relation to commercial arith- 
metic, but because the variety and complexity of the systems 
of weights and measures then in use laid a heavy burden on 
methods of calculation. This is well illustrated in a work by 
Cataneo. 1 In order to make his application of denominate 
numbers clear, it is necessary to explain a few tables : 

Measures of Length. 
12 momementi make one minuto 
12 minuti make one atomo 
12 atomi make one punto 
12 punti make one oncia 
12 oncie make one braccia 
(6 braccie make one cauezza). 
The braccia was equivalent to 31 inches. 

Measures of Surface. 
1 cauezza by 1 cauezza is an area of J4 tauole, or 3 piedi 
1 cauezza by 1 braccia is an area of */> piede 
1 cauezza by 1 oncia is an area of Yz oncia 
1 cauezza by 1 punto is an area of ^ punto 

1 braccia by 1 braccia is an area of 1 oncia 
1 braccia by 1 oncia is an area of 1 punto 
1 braccia by 1 punto is an area of 1 atomo 

1 Cataneo, Dell' arte Del Misvrare Libri Dve. 

12, mjomementi fanno vn minuto. 

12, minuti, fanno vn atomo. 

12, atomi, fanno vn punto. 

12, punti, fanno vn oncia. 

12, oncie, fanno vn piede, in fuperficie, & vn braccia in linea. 

Fol. C 3 verso. 
Cauezzi fia cauezzi, fanno quarti di tauole, ouero piede 3 fuperficiali. 
Cauezzi fia braccie, fanno mezi piedi fuperficiali. 
Cauezzi fia oncie, fanno meze oncie fuperficiali. 
Cauezzi fia punti, fanno mezi punti fuperficiali. 
Braccia fia braccia, fanno oncie fuperficiali. 
Braccia fia oncie, fanno punti fuperficiali. 
Braccia fia punti, fanno atomi fuperficiali. 
Oncie fia oncie, fanno atomi fuperficiali. 
Oncie fia punti, fanno minuti fuperficiali. 
Punti fia punti, fanno moimenti fuperficiali. Fol. C 3 recto and verso. ' 



THE ESSENTIAL FEATURES 79 

1 oncia .by 1 oncia is an area of 1 atomo 
1 oncia by 1 punto is an area of 1 minuto 

1 punto ib}' 1 punto is an area of 1 momenta 

The ratio between the consecutive square units is 12. That is, 12 mom. 
= 1 min., 12 imin. = 1 atom, 12 at. = 1 punti, .etc. 

The following is Cataneo's method of computing the area 
of the trapezoid whose dimensions are given in the figure : 

Testa cauezzi 17, 2, 9 




Testa cauezzi 10, 

" Settima Ragione, Delia (The seventh solution of the) 

quints Figvra (fifth figure = one above) 

(Upper base) Tefta cau. 17, bra. 2, on. 9. ) (Lengths of bases as given 



: 



(Lower base) Tefta cau. 19, bra. 5, on. 8. j in the figure.) 
(Sum) Sornma cau. 37, bra. 2, on. 5. (See linear table.) 



1 Larghezza cau. 18, bra. 4, on, 2, pun. 6. (Half sum of bases.) 
Lunghezza cau. 22, bra. 4, on. 9. (Altitude.) 



2 Doppi cauezzi 9, bra. 4, on. 2, pun. 6. 
Doppi cauezzi 11, bra. 4, on. 9. 



Tauole 


99. 










Tauole 


3, 


pie 


8. 






Tauole 


0, 


pie 


1, 


on. 


10. 


Tauole 


0, 


pie 


0, 


on. 


5, pun. 6. 


Tauole 


3, 


pie 


0. 






Tauole 


0, 


pie 


1, 


on. 


4- 


Tauole 


o, 


pie 


0, 


on. 


0, pun. 8. 


Tauole 


o> 


pie 


0, 


on. 


0, pun. 2. 


Tauole 


0, 


pie 


6, 


on. 


9- 


Tauole 


0, 


pie 


0, 


on. 


3- 


Tauole 


0, 


pie 


0, 


on. 


0, pun. 1, at. 6. 


Tauole 


o, 


pie 


0, 


on. 


0, pun. 0, at. 4, m. 6. 


Tauole 


106. 


pie 


6. 


on. 


8. pun. 5. at. 10. m. 6. 






pun 2 


i|6 min. 


Proua 


4 


Dncie 3 


,|6 min. Fol. G 4 recto. 



1 Larghezza is width and Lunghezza is length. The area of the trape- 
zoid equals that of a rectangle whose .dimensions are J /2 the sum of the 
Ijases of the trapezoid and its altitude. 

2 Since 1 cau. X 1 cau. = V± Tauole (see tables), Yi of 18 and V2 of 



80 SIXTEENTH CENTURY ARITHMETIC 

The vigorous commercial activity of that time demanded a 
knowledge of weights and measures used in all the trading 
centers of Europe. A comparison of these reveals not only 
a great number of denominations, but also a lack of uniform- 
ity in each denomination. An idea of what a " hundred- 
weight " might mean in the fifteenth century may be obtained 
from this excerpt from Chiarini: * "The ioo lb. of Florence 
is 103 lb. in Siena, 102 to 104 in Perugia; in Lucca 102 lb. 
equals 105 lb. in Pisa, and at present is the same as Floren- 
tine weight." 

A comparison of this list with corresponding data given 
by Raets a century later shows the persistency of a condition 
which finally led to the establishment of the International 
System. The following is a typical problem from Raets : 2 
" If a centner of Number g weighs as much as 108 lb. at 
Antwerp, how many centners do 11,682 lb. at Antwerp 
weigh?" In these problems the value of the centner of 
Genoa, Venice, and Antwerp is compared with that of 
Nuremberg, Aquila, Augsburg, England, Bruges, Lisbon, 
Sicily, and other cities and countries. 

An idea of the field covered by the tables of denominate 
numbers required in the practical arithmetic of that time may 
be had from the following summary of Kobel's treatment : 

1. A list of abbreviations of weight and money denom- 
inations. 

2. Tables of money : Rhenish, Frankfurt, Nuremberg, Aus- 

22 are written down before multiplying. Then the result is 99 whole 
tauole. The next step is to find 22 cau. X 4 bra. This is done by finding 
n cau. X 4 bra. = 3 tau. 8 pie. (See tables.) The result is placed as the 
second partial product. When all of the terms of the multiplicand have 
been multiplied by each term of the multiplier, all the results are added 
as shown. 

1 Giorgio Chiarini, Qvesta e ellibro che tracta de Mercatantie et vsanze 
de paesi (1481 ed.). 

" I IBBRE cento Di Firenze fanma in Siena lib o ceto tre i pugia lib. 
cii.I. ciiij. In Lucoa lib. c. ii. I Pifa lib. c.v. & hora e tucto uno conquel 
di Firenze - - -." Fol. 5. 

2 Raets, Een niew Cijfferboeck (1580 ed.), "So den Centner Nuren- 
burghs weecht tot Antwerpen 108 lb. 'hoe veel Centners doe 11682 lb. Ant- 
werp's ?" Fol. Hiiii recto. 



THE ESSENTIAL FEATURES 8 1 

trian, Hungarian, Meissen, Augsburg, Strassburg, Wirten- 
berg, Venetian, Parisian, with comparisons. 1 

3. Tables of common weight. 2 

4. The value of a centner in Venice, Nuremberg, Frank- 
furt, Genoa, Prussia. 

5. Table of gold and silver weight. \ Worms, Oppenheim, 

6. Table of wine measure. \ Mainz. 

7. The number of Omen in a Fuder in Heidelberg, Speier, 
Wachenheim, Durckeim. 

8. Table of grain and fruit measure. 

9. Table of time : minutes, hours, days, weeks, and years. 
He divided the minutes into 18 Puncten instead of into 60 
seconds, and gave 364 days for a year. 

10. Table of cloth measure. 

11. Table of measure of Fustian. 

12. Measure of salt fish. 

Aliquot parts were commonly treated by commercial writ- 
ers. Their importance has never waned, although they have 
often been neglected, and their character changed. Baker 
gave this definition of aliquot parts, : 3 "An aliquot part 

1 Kobel, Zwey rechenbuchlin (1537 ed.). " Der Churfiirften Miintz 
ain Rhein, fol. B ? verso ; Miintz Franckfurter Wehrung, fol. B 8 recto ; 
Miintz zu Nurenbergk, fol. B g recto; Ofterreichifch Miintz, fol. B g verso; 
Vngerif ch Miintz, fol. B g verso ; Meifinif oh Miintz, fol. B g verso ; 
Miintz zu Augfiburgk, fol. B g verso ; Miintz zu StraSburgk, fol. B g verso ; 
Miintz, in Wirtenberger land, fol. C recto ; Der Venediger Miintz, fol. C 
recto; Miintz, zu Pariii." Fol. C verso. 

2 Kobel, Zwey rechenbuchlin (1537 ed.). 

" Von gemeynen Gewichten. 

Centner Qx. 100. lb. 

Pfundt lb 32. lot. 

Ein Lot It hat 4. quintz/ad' ein halbe vntz. 

Quint qui 4. ^ 

Mark mar *26. lot. " 

*26 should be 16. Fol. C 2 verso. 

One centner = 100 lb. 

" pfund = 32 lot 

" lot = 4 quintal, or y 2 ounce 

" Quintal == 4 pfennige 

" Mark = 16 lot 
s Baker, The Well Spring of Sciences (1580 ed.), fol. Mvii verso. 



82 



SIXTEENTH CENTURY ARITHMETIC 



is an eue part of a shiling or of a pound or of any 
other thing*, as %, %, %, %, &c, are called aliquot parts." 
He then discussed the aliquot parts of a shilling so that in the 
reduction the fractions of a shilling may easily be replaced by 
pence. 

Besides the tables of weights and measures there were tables 
to assist in the solution of problems containing denominate 
numbers. An excellent specimen is a work compiled by Jean, 1 
in which the author 
showed how to 
work problems in 
multiplication, 
Rule of Three, In- 
verse Rule of 
Three, and interest. 
His first table com- 
posed of multiples 
of monetary units 
occupies forty-six 
octavo pages. 

The numbers at 
the top of the col- 
umn begin at i and 
proceed to 200,- 
000. The numbers 
in the first column 
begin with 1 and 
proceed to 25. Un- 
der each of the 

column headings there are three divisions for the livre, sou, 
and denier respectively. 

He gave a problem, and explained its solution thus : " Sup- 
pose that 29 aunes of merchandise have been bought at 7 livres 
1 1 sous 9 deniers an aune, to obtain the cost it is necessary to 
find column 29 and go down the column containing lira until 
you are opposite to the 7 of the small tree. Here you will 

1 Alexander Jean, Aritlimetique Av Miroir (1637 ed.), fol. Aij verso. 











) 




fM$JT L-±m* ,1 *°m 


'\hi,r,r 


fcl K 


drnimrffiurej 


Jolz. \d(iucr3{!iiirrj 


./Si- 1&«BT^ 


\ 


TJ 


2.8 


■.3.8. 


•1- 4^ 2 .9 


:zg7 


■■■• 5 Pi 3 


.'30. |"jt. 6 


- * 


5 6 


:s6. 


.•4. 8 


98 


:SB. 


.•4 . 10? f>o 


?•• 


: 5- 


3 | 


84 


4.4. 


=7- 


67 


4:7. 


■'7. Ho" 


4:10. 


:7. « 


4 I 


1 1 X 


S:tX. 


:P- 4 


ti6 


y:,£ 


■'*• 8 J 


1 20 


0: 


:I0 . 


JL 


140 


7: 


:ir. 8 


149 


7: y. 


.12.. 1 


.;< 


7:10 


:i2.d 


% 


168 


8:8. 


•<4- 


174 


8:: 4 . 


:, 4 .tf 


t8o 


9- 


ay. 


1 g 6 


9:16. 


:i6\ 4 


2.PJ 


iff: j. 


no', if jj 3 1 


:o:\o 


«7.* 


9 


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11:4. 


a8. 8 


2. ? a 


i?:.ia. 


'.19. 4I 140 


12: 


:xo. 


j.Sz 


12:12. 


:i/. 


2<m 


rjii. 


:«i.j»j 27° 


13:10 


:xx.6 


to 


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14: 


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:2 4 .aJ2foo 


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:iy. 8 


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1 y.n?. 


3^hy« 


x6\w 


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3? 6 


idm£|:&8. 8748 


17:8. 


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18: 


:?o. 




'3 


3 6 4 


.8: 4- 


■■■5°-4\377 


1 8- '7- 


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1^ 10 


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i<?:ra. 


iV-Z\4°* 


zero". |:J5J0 


420 


21: • 


■■is- 




111 


4Z0 , U«_ 


ll? : J4JL£. 


* '■'?• 


v^.jjU yo 


2 .?.'.!£ |:5 7. o"' 


i 


iB 


^8\2t:8. 


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24:17.(141. ij 710 


2f.VC \-42.6 






M 


yoj Jzy.-j^s,. jfaa. 


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732. 


2<5::2.j.^4.4»5- ? j 


2.7:11. 


■AS-'Asio 


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1 




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ftf 


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5 8 6 


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THE ESSENTIAL FEATURES 83 

find 203, the number of livres for the result." 1 He finds the 
other products in the same way and combines them to find 
the result. 

If one were to pass from the field of arithmetical text-books 
and aim at completeness in describing the denominate number 
systems, the result would be voluminous. An excellent idea 
of the arithmetic of the custom-houses of that time is given 
by Bartholomeo di Pasi. His work 2 of 200 octavo pages is a 
compilation of tables mostly of this kind : 

(The values of the lira in various cities.) 



Melano 


L 239 


Bolzano L 239 


Firenza 


227 


Parma 140 


Genoa 


247 


Nolimbergo 156 


Bologna 


2i6K> 


Geneura 164 


Roma 


217 


Auignone 185 


Napoli 


244 


Parife 179 


Piafenza 


239 


Lione 182 


Mantoa 


239 


Marfiglia 193 


Ferrara 


226 


Valenza 215 



Fol. D g verso. 

The values of the lira, the pezza (measure of length) and 
others are given for over a hundred cities. 

Writers of commercial arithmetic, on account of their ten- 
dency to emphasize the utility of the processes, generally 
placed denominate-number problems under each operation with 
integers instead of deferring the whole subject to a separate 
chapter. Baker, after having explained the addition of in- 

1 Jean, Arithmetique Av Miroir (1637 ed.). '' Suppofez avoir achepte 
29 aunes de marchandife a 7 liures 11 fols. 9 deniers l'aune, il faut trouuer 
la Colomne 29. & defcendre dans les liures d'icelle iufques vis a vis du 7 
du petit arbre, vous y trouuerez 203. qui sont liures. 

Pour les 11. fols, il fault defcendre dans les fols de la dite Colomne 
iufques vis a vis de 11. dudit petit arbre, ou vous trouuerez 15 liures 19 fols. 

Et pour les 9 denier, il faut defcendre dans les deniers d'icelle colomne 
iufques vis a vis du 9. du petit arbre, ou vous trouuerez 21 fol 9 deniers. 

Lesquelles trois fommes fcauoir pour les liures 203 liures, pour les fols 
15 liures 19 fols, & pour les deniers 21 fols 9 deniers, il faut assembler & 
vous trouuerez que 29 aulnes a 7 liures 11 fols 9 deniers, valient 220 liures 
9 deniers." Fol. Aij verso. 

2 Bartholomeo Di Pasi da Vinetia (1557 ed.) Tariffa de i pesi, e misure 
corrispondenti dal Leuante al Ponente, e da una terra, e luogo all' altro, 
quafi per tutte le parti del mondo. 



132 


13 


8 


3456 


16 


5 


789 n. 


17 gr. 


7 d. 


67 


9 


6 


282 


20 


3 



84 SIXTEENTH CENTURY ARITHMETIC 

tegers, passed at once to the addition of pounds, shillings and 
pence. Adam Riese, by common consent the greatest reckon- 
ing master of his time, in his book on line-reckoning began 
addition and subtraction with examples of denominate num- 
bers. Thus, in addition he used the an- 
nexed problem. 1 Even writers who were 
chiefly concerned with traditional arith- 
metic felt the demand for work in de- 
nominate numbers. For instance, under 

division Cirvelo says that division is .„__ TA e 

j 4729 14 5 

used to reduce money of smaller denom- 
ination to larger, just as multiplication is used to re- 
duce money of larger denomination to smaller. 2 Monetary 
systems generally took precedence over all others in order 
of treatment, as indicated by the examples above. In excep- 
tional cases weight was placed first. The plan of introducing 
denominate numbers under addition of integers leads at once 
to a difficulty in sequence of processes. The reductions from 
one denomination to another in simplifying the result occa- 
sionally required a knowledge of division, as in these examples : 



340 flo. 


7 P 25 d 3 




lib 7974 


pl3 


7 4 


124 


7 20 




lib 879 


p 12 


6 


98 


6 27 




lib 9400 


P 5 


7 


49 


12 




lib 794 


P 8 


9 


_58 


6 18 


Suma 


lib 19049 


P 


5 


672 flo. 


5 12 d 





As to the general character of the exercises given under the 
subject of denominate numbers, it is worthy of note that they 

1 Riese, Recihnung auff der Linien und Federn/ (1571 ed.), fol. Aiiij 
recto. 

2 Cirvelo, Tractatus Arithmetice (1513 ed.). 

"CE Finis diuifionis eft vt fciamus quo mo pluribus debet diftribui aliqua 
peeunia fecundu partes equales quantu debet habere e quisq } eorum, et qn 
habemus aliqua magna copia denariorum et voluerimus videre quot folidi 
vel quot aurei aut argentei fierent ex illis et ad plura alia valet, vnde ficut 
multiplioationem poffumus groffiorem reducere ad f ubtiliorem : ita per 
diuifionem poffumus ex minore moneta conftituere maiorem." Fol. a vii 
recto. 

3 Rudolff, Kunstliche rechnung (i534 ed.), fol. Bvi verso. 

4 Cardan, Practica Arithmetice (i539 ed-), fol. Avi recto. 



THE ESSENTIAL FEATURES 



85 



did not contain long lists of numbers to be manipulated merely 
for practice in figuring. Concrete applications were plenti- 
fully supplied, and ability on the part of the learner to solve 
these practical problems was evidently the goal oi instruction. 

' FRACTIONS 

Definitions 
Two conceptions of the fraction were prevalent among the 
writers of that period : ( 1 ) A fraction is one or more of the 
equal parts of a unit. (2) A fraction is the indicated quotient 
of two integers. Among the examples of the former is the 
treatment by Kobel, which reminds us how long the worthy 
apple has done educational service. 1 He divides the apple into 
twenty parts, each part of which is called a twentieth. Ten of 
these parts make half the apple, and five of the twenty parts 
taken together make a quarter of the apple, and so on. Kobel 
is unique in opening the subject of fractions with a statement 
of their utility : 2 " Since it happens that commercial questions 
concerning measure, weight, and exchange are not always 
asked and reckoned with in whole numbers, I shall instruct 
you in the following pages, so that you may understand how 
to arrange, interpret, and reckon questions involving calcula- 
tion with fractions occurring in measure and weight. I prom- 

1 Kobel, Zwey rechenbuchlin (1537 ed.). "So du ein gantzen apffel 
East/ vnd zerfchneideft oder theyleft den felben/ in zwentzig teyl oder 
ftuck/ so ift der felben zwentzig theyl odder ftuck/ ieglichs ein zwentzigft 
theyl des gantzenn apffels genant/ vnnd wirt inn der rechenfchafft iedes 
ftuck ein bruch geteutfcht/ der felben zwentzigtheyl/ zehen/ fo man die 
widerumb zufamen fetzt/ zeygen fie ein halben apffel an/ vnnd fo du der 
zweintzigtheyl fiinff zufamen legft/ fiheftu ein viertheil des apffels z c - 
vnd alfo fiir vnd fiir zu rechnen fein die theyl oder briich zuuenftehn." 
Fol. H 3 verso. 

2 Kobel, Zwey rechenbuchlin (1537 ed.)- " Dieweil fich nit allweg 
begibt/ das die handel/ kauff und fragen/ in gantzen zalen/ maffen/ 
gewichten/ oder verwechfilungenn gefchehen/ gefragt vnnd gerechnet wer- 
den/ wil ich ddch hernach leren/ fo dir in fragenn/ oder rechnungen 
gebrochne zalen/ ungerade gelt/ mafi oder gewicht fiirkumpt/ wie du das 
ordnen/ verftehn und rechnen folt/ fo vil zu difem gmeynem heufilichem 
gebrauah und Rechnen/ ich dir verheyffen ufi einem angenden Rechner am 
erften zu wiffen not ift offenbaren. , ' Fol. H, recto. 



86 SIXTEENTH CENTURY ARITHMETIC 

ise to teach you as much as is evidently necessary for every- 
day use for a beginner in the art of calculation." 

Another example of the first definition, and one in which 
the measured units of denominate numbers serve to define the 
fraction, is given by Champenois as follows : 1 "A fraction is 
part of an integral whole. As a livre is an integral whole, 
and its parts are 20 sous; and one sou is an integral whole 
whose parts are 12 deniers; an aune is an integral whole, 
and its parts are three tiers, four quarts, and other parts, 
a c d e b 



" If the aune ab be divided into four equal parts at the 
points c, d, and e, acde will be the three-fourths, which the 
purchaser took, and the other fourth, eb, will be kept by the 
merchant." 2 The graphical method of explaining fractions 
was very rare in that period. 

The definition of Gemma Frisius is of the same kind and 
contains an explanation of the terms numerator and denomi- 
nator. 3 " We call the numbers showing the parts of an in- 
tegral thing fractions, or parts, as ^ signifies one-half; Ya, a 

1 Champenois, Les Institvtions De L'Arithmetique (1578 ed.). "Frac- 
tion eft partie d'vn entier. Comme vne liure eft vn entier, & fes par- 
ties font 20. fols. & vn fols eft vn entier, & fes parties font 12 deniers, 
vne aulne eft vn entier, & fes parties Tot trois tiers, quatre quarts, & autres 
parties." Page 85, fol. Giij recto. 

2 Champenois, Les Institvtions De L'Arithmetique (1578 ed.). " Soit 
1'aul.ne .a.ib. diuifee en quatre parties egales, au poinct .c.d.e. les trois quarts 
feront a.c.d.e. que 1'achepteur predra, & reftera ,1'autre quart .au marchant 
.e.b." Page 86, Giij verso. 

3 Gemma Frisius, Arithmeticae Practicae Methodus Facilis (1581 ed.). 
" Fractiones, minutias, aut partes, appellamus numeros integrae rei par- 
tes significantes, vt -J semissem. significat, J quadrantem siue quartam 
partem, ^ dodrantem, aut tres quadrantes. Scribuntur duobus numeris, 
superiorem numeratorem, inferiorem denominatorem appellant: hunc quod 
denotet, quot in partes integrum secari oporteat : ilium, quia quot huius- 
modi sumendae sint particulae, numeret. Veluti A, hie inferior denotat 
integrum dividendum in 7, sumendas tamen tantu itres septimas innuit 
superior. Cum igitur duo hi fuerint aequales, semper integrum tamtum >de- 
notatur, vt 1|-. Cum superior maior est, plus integro : cum minor est, 
minus integro significat." Fol. C 2 verso. 



THE ESSENTIAL FEATURES 



87 



fourth; and Ya, three- fourths. They are written with two 
numbers ; the upper one is called the numerator, the lower one 
the denominator, the latter of which denotes into how many 
parts the integer must be divided, the former shows how many 
of these parts are to be taken. For example, in \ the lower 
number denotes that the integer is to be divided into seven 
parts ; the upper one shows, however, that only three-sevenths 
are to be taken. Therefore, when these two numbers arc 
equal, a whole number is designated, as \\. When the upper 
number is greater, it signifies more than a whole number; 
when it is less, it signifies less than a whole number." 

The second conception of the fraction is well illustrated 
. from Trenchant : 1 " The teaching of fractions, of which we 
have given the definition in Chapter 2, should follow division, 
so as to follow the proper source from which it originates. 
For this will happen most often when a smaller number is to 
be divided by a larger ; or when it results from a division, as 
24 divided by 60 makes ff ; or when it results from a divi- 
sion, as 24 resulting from a division by 60 makes f|. The 
24 is the numerator and 60 the denominator, and the fraction 
is called twenty-four sixtieths." 

The second form 1 of definition was used by Raets : 2 " Frac- 
tions arise (as has been explained) from division of a number 
by a greater number. For instance, when 2 is divided by 3, 
then f results," and also by Tartaglia, who gave this illus- 
tration : 3 " To divide 15 by 2. It will be impossible to divide 
15 into two equal parts. After dividing there will be one 

1 Trenchant, L'Arithmetique (1578 ed.). "La doctrine du nobre rompu, 
duquel auons donne la diffinition au 2 chap, doit fucceder a la diui- 
fion, come fuyuant sa propre fourfe >dont il prent origine. Car iceluy 
auient le plus fouuent quand Ion diuife vn moindre nombre, par vn niaieur : 
comme 24 par 60, fet |~i: on quand il refte d'vne partition: comme 24 
reftant d'vne partition par 60 fet -||-: le 24 eft numerateur : & le 60, 
■denominateur : & s'exprime vin<t & quatre roixantiemes." Fol. G recto. 

2 Raets, Arithmetka Oft Een niew cij fferboeck/ (1580). "Die ghebroken 
ghetalen spruyten (als verclart is) wuter Diuision/ alsmen een ghetal 
divideert met een grooter. Ghrlijck alsmen divideert 2. met 3. Too comen- 
der |-." Fol. Biiij verso. 

3 Tartaglia, La Prima Parte Del General Trattato (1556 ed.). 



88 SIXTEENTH CENTURY ARITHMETIC 

part left, which is still to be divided by the divisor. Then i 
is taken for the numerator and 2 for the denominator of the 
fraction." 

Thus, the proper relation of the fraction to the process of 
division was recognized, and the fraction was taught as a 
necessary step in the growth of the number system. 

The second method of approach to the fraction, though less 
common than the first, persisted for two centuries. It is in- 
teresting to compare Tartaglia's treatment quoted above with 
the following from Gio (1689) more than a century and a 
quarter later : x " When a divisor is greater than a dividend. 
When' it is necessary to divide a smaller dividend by a divisor, 
place the divisor under the dividend, and, as no quotient is 
obtained, a fraction is formed. Thus, if the divisor is 40 and 
the dividend 20, place 40 below the 20, then the quotient £& 
results." 

The degree to which these conceptions of the fraction were 
held to be incompatible by some is exemplified in the work of 
Unicorn, 2 who^ began with the first definition and ended a long 
treatment with these cases of division, the last and least of 
which is the second definition. 
Division of fractions : 

A fraction by a fraction. 

An integer by a fraction. 

A mixed number by a fraction. 

A mixed number by a mixed number. 

A fraction by an integer. 

An integer by a mixed number. 

A fraction by a mixed number. 

A mixed number by an integer. 

1 Gio, Padre, Elementi Arithmetici (1689 ed.). Quando un Partitore 
foffe moggiore de composto. Quanto s'haueffe da partire vn Compofto 
minore del Partitore ; si mette il Partitore lotto il compofto, e ne viene di 
Quotients quel rotto die fi forma. Gome se foffe Partitore 40, Composto 
20. Perche il 40. non puo entrare in 20., percio neffo 40. lotto il 20, resta 
di Quotiente |o.. 40)-|£ Quotiente." Fol. Cviij verso. 

2 Unicorn, De L'Arithmetica universale (1508 ed.). 



THE ESSENTIAL FEATURES %g 

One integer by another ; that is, in case the dividend is 
smaller than the divisor, as 48 by 64 gives |f . 

A curious mixture of the two definitions is found in 
"Ramus i 1 " If 8 is divided by 3, the quotient is 2, and % are 
left. 2 is the number of parts, 3 the name. 

If I wish to divide 11 asses by 3, this division ^ (3-f) 
will indicate 3 asses and % of 1 ass. 

If the numerator and the denominator are equal, as fft- 
the number is an integer; if larger, as Iff, it will be more 
than an integer." 

Both were happily combined by Tonstall, whose definition 
is : 2 "Any integer may be divided into as many parts as one 
wishes from 1 1 to infinity." Although Tonstall often goes to 
extremes in details, his treatment of the relative size of frac- 
tions should have a mission. One of the chief reasons why 
children have difficulty in mastering fractions is that they do 
not make a sufficient number of comparisons. They do not 
observe the change in the fraction by varying its terms. Ton- 
stairs comparisons are suggestive of good method : 3 " The 
greater the denominator and the fewer of these parts there are, 
the farther is the fraction removed from the integer; the 
smaller the denominator and the more of these parts there are, 

1 Ramus, Arithmeticae Libri duo (1586 ed.). "Ut efto 8. dividendus 
per 3. quotus integer est 2, & fuperfunt 2, que interpofita linea fupernotata 
divisori, tandem ipfa quoq } divifa funt, invenitaqj fraotio 2 /z quoto priori 2 
ad dextram est af f cribenda lie : 

2 

3 " Fol. D ? recto. 

2 Tonstall, De Arte Supputandi (1522 ed.). " Omne integrum in Partes, 
<juotcuqj velis : folui per intellectu poteft. et quemadmodum integrorum 
numeratio ab uno incipit: atq? in infinitum poteft extendi: Tic integrorum 
lectio a fecudis orditur partibus." Fol. 4 recto. 

3 Tonstall, De Arte Supputandi (1522 ed.). "Semper autem in omnibus 
diffectorum minutijs, quo maior denominator fuerit: eo minores erunt 
partes, remotioresq? ab integris. et quo fuerit minor : eo partes maiores 
•erunt. atq3 ad Integra propius accedet. Nam du§ partes fecundae maiores 

funt : q duae tertig et duae tertiae maiores, q duae quartae et duae 
quarts maiores, q duae quintae. et Tic in uniuerfum, quanto magis numer- 
ado crefcit denominator : tanto magis quantitate partes diminuunter." Fol. 
P recto. 



go SIXTEENTH CENTURY ARITHMETIC 

the more nearly a fraction approaches an integer. For 2 
halves are greater than f, f than J, f than f, and so on. 

" In dealing with fractions these things should be noticed : 

First. Whenever the numerator and denominator are 
equal, then the fraction equals an integer. Thus, f, £ , \ are 
equal to 1. 

Second. Whenever the numerator is greater than the 
denominator, by as many units as the numerator exceeds the 
denominator, the value of the fraction exceeds an integer, as in 
i> h f > t is an integer and }i over, and so> on. 

Third. By as many parts as the numerator is less than 
the denominator in units, by so much is the fraction less than 
an integer, % is one- fourth less than an integer." 

Classes of Fractions 

Two classes of fractions were recognized : 

Common fractions, variously called numeri rotti, fragmenta, 
nombres rouptz, fractiones, 1 minutiae vulgares seu mercatoriae. 

Sexagesimal fractions, called fractiones astronomicae, or 
minutse phisicse. 2 

Cirvelo, in the third part of his Practica Arithmetica (1555 
ed.), explained the operations with the sexagesimal fractions 
by reference to those with denominations of weight and value. 
The following shows his method of associating the addition 
of signs, degrees, minutes, etc., by reference to the addition of 
ducats, soldi, denarii, etc. 

tertia 
15 
34 
45 

11 31 57 57 45 ffia 

1 The word " fractio " is as old as Hispalensis (c. 1150). Treutlein, Ab- 
handlungen, 3 : 112. 

2 Planu'des introduced sexagesimal fractions under the title " Zodiac "" 
and showed how to use them in the four operations. According to Sayce 
and Bosanquet, the origin of these fractions is sometimes incorrectly attrib- 
uted to the Assyrians. Publications of the Royal Asiatic Society, 1880, vol. 
xl, no. 3. 



sign a 


gradus 


minuta 


secur 


5 


47 


39 


53 


3 


26 


54 


18 


2 


17 


23 


45 



THE ESSENTIAL FEATURES 



91 







Practica 


IN 


MONETIS. 




Auri 
dticati 


argenti 
solidi 


denarii 




oboli 




12 

8 

23 


23 
16 

14 


9 

7 

11 




4 
3 

5 




53 
6 oboli = 


25 
1 denarius. 


4 
12 denarii 




fua 
1 solidus. < 


Fol. b 2ij recto. 
50 solidi = 1 due; 



A double entry multiplication table is also given by Cirvelo, 
page 38, but it was not an invention of the sixteenth century, 
for the Arabs had used this means of calculation much earlier. 
In German works that treated of line-reckoning, Roman 
notation was occasionally used to express fractions. An ex- 
cellent illustration showing the struggle of the Hindu and 
Roman systems is the following from Kobel : * 

" The numerator 1 This symbol is one-oneth, 

The separatrix — that is, the integer 1. 

The denominator 1 
" Then, whatever equal numbers are found in the numerator 
and denominator, they always mean in such a symbol the in- 
teger 1 ; for example, 4 fourths are 1 and written J4n i n th e 
German numerals and -f in figures. Similarly, 6 sixths is 
also 1 and written ^ or f . " The illustration on the next page 
shows a part of a page from' Kobel's arithmetic (1544). 

1 Kobel, Zwey rechenbuchlin (1537 ed.). 

" Der zeler I DiS figur ift vnd bedeut ein eintheyl/ das 

Das ftrichlin — ift ein gantzs. 

Der Nenner I 
Dann in welcher zal du den Zeler vnd Nefier gleich findeft/ fo bedeutten 
die felbigen figuren .alwegen ein gantze zal/ <als vier viertheil/ die machen 
ein gantz/ vnd werden alio mit teutfeher zal gefchriben jijj/ aber mit 
den ziffern alio ^ Dergleichen ift 6. fechfteil auch ein gantzs/ vfi fchreib 
es alio yj/ vnd mit den ziffern -|." Fol. H 4 recto. For examples from 
other of Kobel's works see Unger, pp. 15-16. 



9 2 



SIXTEENTH CENTURY ARITHMETIC 



bcbtJk biftfis** bet fetbm bttf ab» • 

I fcfcfle ftgur if! t)it befcefft aw* jfertel Don tin $ 
IUT jyarojeiVaf fo mag matt <m<g> airs fifo fftail/a^ 
fedjftou/ainftibeMtil mrswtife<fyftmzc*yrib alls 
M6cr br8c^ befc^mbe»/2«0 7 1 ^ I yjr I $• 2c. 

VF SD$ fern &cdre acfrail/fcae fcin fedjetatl bw 
VJIT 6d)t am $mb m&fym • 

IX &if?£ig»r Begafgt aim newtt ayilfjtatffcasfeyii 

XX &$j?igt>r bet3at$et/$wmi3tgfc aftfimfcrey* 
XXXI figfc Mil /fras few jjw wijfgfc tail *&er aiite* 
Httfcro'jfigfc rtt«ga«©mac|je«4 

II C '&$ feiJi tzw&ifyunbert Mil/fcer jfier&im'* 
III1ALX frert mb fcdtBigt m gm% matyen # 



Many writers followed their definition of fractions by an 
explanation of numeration of fractions. This table from Van 
der Schuere was designed to teach numeration to yV* * 



l 


g 

2 


a 
s 


4 
4 


6 


6 


7 
7 


a 

8 


a 

9 




1 
2 


2 
3 


a 

4 


4 
6 


5 
6 


6 

7~ 


7 
8 


8 

9 






1 

3 


4 


a 

5 


6 


7 


8 


7 

9 






L 

4 


a 

5 


a 

6 


7 


8 


9 






1 
6 


a 

6 


a 

7 


4. 

8 


9 






1 
6 


s 

7 


a 

8 


1 

9 






I 

7 


8 


a 

9 






1 
8 


9 






l 

9 



1 Van der Scheure, Arithmetica, Oft Reken=const (1600 ed.). Fol. D g 
verso. 



THE ESSENTIAL FEATURES 93 

Order of Processes 

The order of the processes with integers seems to have im- 
pressed itself so generally upon writers that it was usually fol- 
lowed blindly in the case of fractions. The order of presenta- 
tion in fractions now prevalent in formal arithmetic was the 
one commonly used in the sixteenth century, namely: 1. De- 
finition. 2. Reduction to lower terms. 3. Reduction to' the 
same denominator. 4. Addition. 5. Subtraction. 6. Mul- 
tiplication. 7. Division. 

The following outline from Tartaglia is typical as to the or- 
der of the main topics ; 1 the order and number of minor topics 
differed with different authors : 

Numeration, or representation of fractions. 
Reduction of fractions to lower terms. 
Changing mixed numbers to fractions. 
Reduction to the same denominator. 
Addition. 

Of fractions. 

Of fractions and mixed numbers. 

Of fractions, whole and mixed number. 
Subtraction. 

Of fractions. 

Of fractions from mixed numbers. 

Of fractions from whole numbers. 

Of whole numbers from mixed numbers. 

Of mixed numbers from mixed numbers. 
Multiplication. 

Of a fraction by a fraction. 

Of a mixed number by a fraction^ and conversely. 

Of a fraction by a whole number. 

Of mixed numbers by mixed numbers. 

Of whole numbers by fractions. 

Of a fraction by a fraction by a fraction. 

Of a whole number by a mixed number by a mixed number. 
Division. 

Of a fraction by a fraction of the same denominator. 

Of a fraction by a fraction of different denominator. 

Of a whole number by a fraction. 

Of a mixed number by a fraction. 

Of a mixed number by a mixed number. 

Of a whole number by a mixed number. 

Of >a mixed number by a whole number. 

1 Tartaglia, La Prima Parte Del General Trattato di numeri, et misure 
(1556 ed.). 



94 



SIXTEENTH CENTURY ARITHMETIC 



But should the order of processes with fractions in formal 
arithmetic necessarily be the same as that with integers ? It 
is not followed in primary arithmetic, it does not conform 
readily to spiral treatment, and, furthermore, historical pre- 
cedent is not unanimous in its favor. Calandri (1491), Borgi 
{1484), Paciuolo (1494), and Rudolf! (1526) each gave 
the following order : multiplication, addition, subtraction, and 
division. Gemma Frisius introduced multiplication first and 
repeated it after subtraction. The subject was introduced by 
the definition of a fraction and its terms, and followed by the 
explanation of a fraction of a fraction. 1 "As | of f of f , 
that is, the integer is divided into 7 parts of which 6 are taken, 
this is again divided into 3 parts, of which 2 are taken, and the 
resulting fraction is divided into 4 parts, of which 3 are taken. 
The fractions are most easily multiplied by multiplying the nu- 
merators together for a new numerator and the denominators 
together for a new denominator." Francesco Ghaligai pre- 
ceded addition and subtraction by both multiplication and 
division. 2 These presentations cannot be regarded as experi- 
ments of minor authors, for Paciuolo, Gemma Frisius, and 
Calandri were representatives of the highest scholarship and 
were leading spirits of both the Latin and the Commercial 
Schools. Neither can they be regarded as entirely arbitrary or 
accidental plans, for Paciuolo, who did more than any other 
writer (save possibly Borgi) to formulate the arithmetic of 
that period, actually expressed his purpose thus : 3 "In the ex- 

1 Gemma Frisius, Arithmeticae Practicae Methodus Facilis (1581 ed.). 
" Item |. I |. J 6.^ hoc est, tres quare a >duarum tertiaru ex lex feptimis : 
hoc eft integri diuifi in 7, cape lex particulas : quas rurfus feca in tres: 
harum accipe duos: quas diuide in quatuor: tandem tres huiufmodi fig- 
nincatur particule." Fol. C 3 recto. 

2 Ghaligai, Practica D'Arithmetica (1552 ed.). 

3 Paciuolo, Suma de Arithmetica Geometria Proportioni et Proportion- 
alita (1523 ed.). 

" De multiplicatione fractorum inter se. 

- - - Nelli fani el lummare : fe ben notafti fo piu facile che nullaltra a 
parte: el multiplicare fo piu difficile che lui. Qui nelli rotti asfai piu 
facile : e k> multiplicare che non e lo f ummare. E nelli fani la piu difficile 
parte fo el partire. E qui nelli rotti e la piu facile : che quella intendi di 



THE ESSENTIAL FEATURES 95 

planation of reckoning with fractions, multiplication precedes 
addition, for fractions and whole numbers are so different that 
in whole numbers addition is more easily treated, while in 
fractions it is more difficult than multiplication and should 
follow." 

Reduction 
The reduction of fractions to lowest terms was effected in 
simple cases by inspection. In other cases it was necessary to 
apply tests of divisibility to find the common factors of the 
numerator and the denominator. Reduction to lowest terms 
was the only method of simplifying fractions, since the decimal 
fractions had not yet come into use; consequently, the tests of 
divisibility, and occasionally the Euclidean method of Great- 
est Cornmon Divisor, were given by theoretic writers before 
fractions. For example, Ramus preceded his treatment of 
fractions by five chapters, as follows : x 

Chapter VI. Concerning odd and even numbers. 

From division arise different kinds of num- 
bers : odd and even, prime and composite. 
Chapter VII. Concerning prime and composite numbers, 
in which the sieve of Eratosthenes is used. 
Chapter VIII. Concerning numbers prime to one another. 
Chapter IX. Concerning composite numbers and their 

Greatest Common Divisor. 
Chapter X. Concerning the Least Common Denomin- 

ator. 

fani. El fotrare in quelli ene asfai piu facile: idle non e el fotrare in li 
rotti. E pero dal multiplicare me pare ben eomenzare." Fol. 50 recto, Gij 
recto. 
Recorde (1558 ed.) makes the same explanation. Fol. Riiij verso. 

1 Ramus, Arithmeticae Libri duo (1586 ed.). 
" Caput VI De Numero Impari et Pari. 

E divifione oritur numeri differentia duplex, imparis & paris, primis & 
■compofiti. Fol. C 5 recto. 

Caput VII. De Numero Primo et Composite Fol. C G recto. 

Caput VIII. De Numeris Primis inter Se. 

Cap. IX. De Numeris Inter Se compofitis, eorumq; communi divi- 

for-e maximo. Fol. D g verso. 
Cap. X. De minimo Communi Dividuo." Fol. D 4 verso. 



q6 SIXTEENTH CENTURY ARITHMETIC 

Stevinus under division of integers states the use to be made 
of the Greatest Common Divisor; namely, as an aid in the re- 
duction of fractions. 1 It was more common, especially among 
commercial writers, to give the tests of divisibility under frac- 
tions or in connection with the actual problem solved. Thus, 
in RudolfFs second section in the treatment of fractions Re- 
duction has this treatment : 2 The first example, solved by in- 
spection, — both terms of ^4 divided by 2 = T 4 ^, which 
divided by 3 = if, which divided by 7 = ?, — is followed 
by the rules of divisibility : 

" In order to be divisible by 2 a number must be even ; 

" In order to be divisible by 3 it must have a remainder 
of o, 3, or 6 after the nines have been cast out ; 

" A number is divisible by 5, when the number ends in 
5 or o; 

" A number is divisible by 10, when the number ends in o." 

As an example of these tests he gives the one in the tttt 

margin. Then follows the Euclidean method : 24S 

" How we may easily find whether a fraction may be 

made smaller or not. 2-fr 

" Divide the larger number by the smaller. If any- 2 7 
thing is left, divide the former divisor (the smaller 

number of the fraction) by this, and so on. Then ■$ 

divide the larger number of the fraction, and, if the s 
division comes out even, the fraction is reducible. As, 



715 



286 = 2, remainder 143. 



28| 286 -r- 143 = 2, no remainder. 

715 ~ 143 = 5." 

1 Stevki, Les Oeuvres Mathematiques> (1634 e&). " Ef tant doncques 
donnez nombres Arithmeticques entiers, nous avons trouve leur plus grande 
commune mefure ; ce qu'il f alio it f aire. 

■Problem VI. 

Estant donne nombre Arithmetique rompu : Trouver Ton premier ronipu. 

Explication du donne. Soit donne <rompu -f^j. Explication du requis. 
II faut trouver fon premier rompu. Conftruction. On trouvera la plus 
grande commune mefure de 91 & 117." Fol. B 5 recto. 

2 Rudolff, Kunstliche reclmung mitder Ziffer und Zalpfennige/ (1534 ed.)- 



THE ESSENTIAL FEATURES gy 

Another interesting- example is found in Raets. 1 
The term, " reduction of fractions " meant several things : 
with some writers it meant reduction to lowest terms, with 
others it meant changing fractions to others having a common 
denominator, and with still others it meant expressing a frac- 
tional part of a denominate number unit in terms of lower 
orders. It is now customary for modern writers to precede 
the treatment of common denominators by that of reduction 
to lowest term's, but the influence of applied arithmetic, es- 
pecially of denominate numbers, and the freedom.' accompany- 
ing dogmatic treatment sometimes led sixteenth century writ- 
ers to reverse this order. Thus, Baker's second chapter is 
Reduction of Fractions to Common Denominator, and his. 
third chapter is Reduction of Fractions to Lowest Terms. The 
influence of denominate numbers upon the treatment of frac- 
tions is apparent among both Commercial and Latin School 
writers. Thus, Adam Riese follows his definition of a frac- 
tion by : 2 

To .reduce ^ floren. 

Solution. Multiply 3 by 21 and divide by 4, the result is 15 gr. and 9 <?~ 



1 Raets, Arithmetioa Oft Een niew Ci j fferboeck/ (1580 ed.). 
"Abbreviatie int ghebroken. 

" Leert/ hoe-men die gebroken ghetalen redueeren sal in een min- 144 

der proportie. Het welcke ghesehiet nae der leeringhe Euclidis/ 108 

In der tweeder propohtien des feuenden boers der Elementen/ in 36 

deser voegen. Ten exemplel : Om habbbeuieren \\\ fuibshaheert 72 

den Felder 108. va den Konimer 144. ende daer fullen resteren 36. 36 

die subtrahert van 108. 36 

o 

" Das voor -1-|| unde plaet se van 108. fedt 3. ende voor 144. fedt 

4. Too comender -| vor 4-ir|te meten/ die £ doen loo veel als y-|f-" FoL 
Bv recto. 

2 Riese, Rechnung auff -der Linien und Federn/ (1571 ed.). 

" Wiltu wiffen/ wie viel ein jglicher Bruch in fich behelt/ To refoluir 
den Zeler in fenien werdt/ vnd teil ab mit dem Nenner. Als ^ floren/ 
multiplicir 3 mit 21 gr./ vnd theil ab mit dem Nenner/ als 4/ kommen 15 
gr./ vnd 9 d- Alio dergleichen von Gewichten vnd andern." Fol. D verso. 



9 8 SIXTEENTH CENTURY ARITHMETIC 

Gemma Frisius follows Reduction of Fractions to Lowest 
Terms by : 

"f Joachimi or Thaleri are worth how many grossi?" 
124 grossi = 1 Thaler. - = 10% grossi. 

?= 8 numuh. 

3 

Ramus precedes his Chapter XIIII, 1 Operations with Frac- 
tions, by a chapter, Reduction of Integers and Fractions, in 
which examples of this type occur : 

To reduce 12 asses to uncias ^-. 

To reduce - X T 4 2 4 to an integer, divide 144 by 12. 

Addition 
The usual order in the addition of fractions was : ( 1 ) The 
addition of fractions with denominators alike, (2) Of those 
with denominators unlike. For example, the following prob- 
lems are given by Noviomagus in this order : 2 

(1) f and * and f = ^. 

(2) f . f . Multiply 7 by 3 = 21, 5 by 4 = 20. 

The sum = 41 for numerator and 7 X 4 = 28 for denominator. 



The second kind required reduction to a common denominator, 
which process was treated either in a section previous to addi- 
tion or in connection with the actual problems to be added. 
The number of writers who followed each plan was about 
the same. 

A typical case of reducing to a common denominator is the 
following f rom Trenchant : 3 To reduce f , i, f , J- to a 
common denominator : 
16 18 20 21 The common denominator is written 

„ below, and the new numerators are 

z. A A Z_ 

3468 written above the corresponding frac- 

2 4 tions. 

1 Ramus, Arithmeticae Libri duo (1577 ed.), ft>l. 4 recto; fol. D g recto. 

2 Noviomagus, De Numeris Libri II (1544 ed.). 

2 Trenchant, L'Airithmetiquie (1578 ed.), fol. G ? verso. 



THE ESSENTIAL FEATURES 99 

The usual form of expressing the work of addition is shown 
in this example from Rudolff : x 
8 9 The fractions to be added, I and f, 

2 3 17 appear in the center, the common de- 

- -ma es— nominator, 12, below, and the new 
12 numerators, 8 and 9, above. 

A clever form, of adding fractions, considering the fact that 
signs of operations were not in use, is illustrated by the fol- 
lowing from Tartaglia : 2 

2 3 g The cross indicates how the terms 

asummar -x- 8 are mll ltiplied, but it probably has 

, 17 . . T 5 no connection with the symbol of 

fanno — che sana 1— ... 

12 multiplication. 

So rarely were reasons given for processes that the follow- 
ing explanation by Rudolff of why fractions must be reduced 
before adding is noteworthy : 3 "It is impossible to add 4 
florins and 3 soldi to make either 7 florins or 7 soldi. It is 
also impossible to> add | and J and to get either f- or -f. 
The florins must be reduced to lower denomination, and the 
fractions must also be reduced to a common denominator. 

8 1 nr 15 8 mo U 23 " 

8 5 or 40 40 make 40-. 

The order, 4 or the method of grouping, used when several 
fractions were added was not uniform. The prevailing usage 
was to add the first fraction to the second, then to add this re- 
sult to the third and this result to the fourth, and so on. 5 

1 Rudolff, Kunstliche Rechnung (1534 ed.), fol. Cvii verso. 

2 Tartaglia, La Prima Parte Del General Trattato (1556 ed.), fol. Tiij 
verso. 

3 Rudolff, Kunstliche Rechnung, fol. Di recto. 

4 Baker, The Well Spring of Sciences (1580 ed.), adds 1, f, |, f by 
adding -1 and |-, then -| and -|, combining the results. Fol. I v verso. 

He also directs bow to add a fraction of a fraction to a fraction of a 
fraction by making the multiplications first and then adding the results. 
This bas significance in showing that Baker gave multiplication precedence 
over addition, as is done in the modern convention concerning a series of 
operations. 

5 Riese, Rechnung auff der Linien und Federn/ (1571 ed.), adds |-, ^, 
and # thus : % and # are 1-1 and 4 = 2i|-. " Summir die erf te zween 

e & 3 4 12 o 16 

bruche/ als nemlich/ J vnd j/ werden 1X/ darzu -J/ kommen 2^|. teil." 
Fol. Dij recto. 

LOFC, 



IOO SIXTEENTH CENTURY ARITHMETIC 

Some writers, as Cardan, 1 added in groups and combined the 
results according- to the associative law. The modern method 
of reducing all of the fractions to a common denominator and 
adding their numerators was also recognized. 2 Tonstall 3 
combined the first and the third methods. In adding f, f, 
and i he suggested that the first two be added and that the 
third be added to this result, as : 

t + I = t! and H X f = W-, or that they may be taken as 

f = U, I = U, i + U. The sum is W- 
When the modern method' was used, the product o>f the sev- 
eral denominators was often taken instead of their least cornh 
moo multiple. The following problem from Raets shows this :** 
To reduce f , |, f . 



4 


160 


8 


7 


32 




5 


3 




W?0(14O 


160 


fflfi 


3 






160 


#20(120 


4 


m 







1 




W(128 



1 Cardan, Practica Arithmetice (1539 ed.). " Exemplu volo agregare §,. 
j, A £ agrego p modu dictu Prima duo & fatiut ±-L & reliq duo & fatiut 
|| 5 deinde agrego ^ & 40 & fi^t i^f & sut integri tres & ^ & hoc 
eft facile." Fol. Bii recto. 

2 It was not customary, however, to use the least common denominator. 

3 Tonstall, De Arte Supputandi (1522 ed.). "Si vero plura fuerunt 
firagmeta: uti duae tertij. tres quartae. quatuor quintae. poft duo priora 
fragmenta, ficuM diximus, reducta, iterum .denominator comunis prius in- 
ueftigatus per tertij fragmenti denominatorem multiplicetur : et f urgent 
f exaginta, omnium denominator communis. 2 |. 4. *_7_ 4. ^ ^ 

♦Should be 17 in the original, which shows that the fractions were added. 
"Q) si fcire cupis : quot partes fexagefimae Tint in quouis fragmento 
numeratorem ipfius fragmenti in denominatorem cSmunemi multiplica: 
nempe f exaginta: numerurnqj procreaitu diuide per eiufdem fragmenti de- 
nominatorem. Ita deprehendes in duabus tertij s quadraginta fexagefimas . 
£ || et in tribus quartis quadraginta quinq? . fexagefimas . J |~| 
et in quatuor quitis quadraginta octo fexigefimas." Fol. P 3 verso. 

4 Raets, Arithmetica Oft Een niew Cijfferboeck (1580 ed.), fol. Bvi verso- 
to Bviii recto. 



W(40 


W(20 


#0(32 


ft 3 


W 7 


?? 4 


120 


140 


128 


5 


4 


4 


8 


5 


8 


40 


20 


32 


3 


7 


4 



THE ESSENTIAL FEATURES IOI 

Therefore, the fractions become -ffj, y|#, tt§- 

2. 



3. 



120 140 128 

This problem has an additional interest on account of its three 
solutions. In the first the common denominator is multiplied 
by each numerator, and the result is divided by the correspond- 
ing' denominator. This is evidently the longest method. In 
the second the common denominator is divided by each de- 
nominator, and the results multiplied by the corresponding nu- 
merators. In the third the product of two denominators is 
multiplied by the numerator of the third fraction; this is the 
shortest process. 

Mixed numbers were added in -two ways : ( i ) By adding the 
integers and fractions separately and combining the results. 
(2) By reducing the mixed numbers to improper fractions and 
adding. The following example from Rudolrr" 1 shows a com- 
mon form for arranging the work in adding by the first 
method : 

13J 3 8 10 6 

7| 

12f ill! 

19* 



53£ 12 12 

The following shows the work of adding mixed numbers by 
the use of the second method : 2 

1 Rudolff, Kunstliche reohnurig (1534 'ed-), fol. Dii recto. 

2 Giaodhi, Regole Generali D'Abbaco (1675 ed.). "Del seccmdo modo di 
sominare interi, e rotti. Supponga li per efempio, ©he vno f i trouaff e debi- 
tore cfvn altro «di diuerfe fomme di danari, cioe di lire 16^-, di lir. 4^-, 
■di lir. n£-, di lir. 25^, di lir. 6£-, di lir. 15.7. 12 fimi, e di lir. 4.5.12 iimi, 
si domando di quanta fi douera af criuere in vna Tola partita ; Si ponghino 



102 


SlXl&bN ltL 


LHJS 1 UK1 


' AKUJr 


IMhllL 




16* 


4f HJ 


25f 


6f 


15 T V 


4& 


1A 


i a 9. 


6. 


A 


a 


3 


¥ 


¥ ¥ 

Somma prima 

seconda 

terza 

quarta 

quinta 

sefta 

settima 


1|5 

594 
168 
405 
930 
232 
56i 
159 


58 


w 


ft 



3049 36] 84.1310.^ 

The top row contains the numbers to be added. The third 
row is composed of the same numbers reduced to improper 
fractions. The middle row contains numbers of which the 
numerator of each fraction is to be multiplied to give an equal 
fraction with the denominator 36. The column which fol- 
lows is the sum of these numerators. The result is expressed 
in lira, soldi, and denarii. 

Under addition and other processes with fractions there 
generally were included problems in denominate numbers. 1 

Subtraction 

The order in subtraction was naturally the same as that in 
addition. That is, the subtraction of fractions ( 1 ) with the 

le fomme per ordine, e fi reduchino gl' interi a quella parte, con la quale fi 
trouano copulati, e s'offerui il modo notato nell' antecedente." Fol. D g 
recto. 

1 This problem from Trenchant (L'Arithmetique, 1578 ed.) will illus- 
trate : 

The fractions represent parts of a lira, 
a money denomination, the second col- 
umn contains the equivalent amounts 
expressed in the lower denominations,, 
soldi and denarii. 



* 


iof s 


1 
i 
* 

t 

TV 


13 4 

5 

16 8 

7 6 

11 8 


3^ 


3l. 4f. 2 6 



THE ESSENTIAL FEATURES ^3 

like denominators, (2) with different denominators. Baker, 1 
Gemma Frisius, 2 and Tonstall 3 emphasized the subtraction of 
a fraction from an integer, performing the process in two 
ways : By detaching one from the integer and subtracting the 
fraction from that, and by reducing the integer to an improper 
fraction of the same denominator as the given fraction and 
then subtracting. 

The subtraction of mixed numbers was accomplished in 
two ways : by subtracting the fractions separately, adding one 
to the minuend when necessary, and by reducing both minuend 
and subtrahend to improper fractions. Theorists like Ton- 
stall 4 wno were fond of extreme classification gave three cases 
under mixed numbers: 

1 . Subtraction of a fraction from a mixed number. 

2. Subtraction of an integer from a mixed number. 

3. Subtraction of a mixed number from a mixed number. 
The placing of ( 1) before (2) in this list is an example of the 
illogical order that characterized even the work of careful 
writers of this period. 

1 Baker, The Well iSpring of Sciences (1580 ed.), fol. Ki recto, Kii recto.. 

2 Gemma Frisius, Arithmetical Practicae Methodus Facilis (1581 ed.), 
fol. Cv verso. 

3 Tonstall, De Arte Supputandi (1522 ed.). To subtract -f. from 12, 
take 1 from 12 which leaves 11. 1 = 1 and 1- — f = f • Hence, 12 — 

♦ = «f 

Or, 12 may ibe reduced to sevenths, making 84.. *± — 4 — _8_o^ TI S t 
" Quando minutiae fubducentur ab initegris : f uff ecerit eas <ab uno integro, 
in minutias foluto, fubdueere. et quod tarn de integris q de minutijs ref- 
tabit: totius fubductionis erit reliquum. Veluti fi |- fubtrahendae funt 
a 12. fumamus .1. de .12. et reftabunt .11. ab illo autem uno demptis A, 
reliquentur -|. qu§ copulatae ,cum. 11. reftare faciunt .n|-. tantum 
fupereft : f i |- a. 12. fubducimus. Alij integra minutiarum more, fupra 
lineam notant: cui unitatem subijciunt, ad integra defignanda. Deinde 
quafi minutiae a .minutijs subducendae effent: pofit obliquam numeratorum 
in denominatores .multiplicaltionem, minorem productumi a rnaiore fub- 
ducunt: et fupra lineam notant. cui denominatorem fubdunt. ita. 7. in 12. 
ducta creant .84. et. 4 in .1. ducta faciunt .4. qu§ fubducta ab .84. reliquunt 
-8^-. Ea, fi reducas ad integra: fient .n|-. ita res ad idem recidet. -i X 
1. -8^0." Fol. S 1 recto. Riese (1571 ed.) used this plan, fol. Dii verso. 

4 Tonstall, De Arte Supputandi (1522), fol. S 2 recto. 



IQ 4 SIXTEENTH CENTURY ARITHMETIC 

Among the works which correlate fractions and denominate 
numbers, it would be difficult to find a better specimen than that 
of Wencelaus. This is an example under subtraction of frac- 
tions: 1 "A silversmith had an amount of silver weighing 15 
marc if once, he wished to make from it an article weighing 
8 marc 5 J once. The question is : How much silver is left ?" 

Multiplication 

The multiplication of fractions was generally based upon the 
definition of a fraction. This statement, often included in the 
general definition O'f a fraction, gave the common rule for 
forming the product of twoi fractions. That the multiplication 
of one fraction by another does not require special emphasis 
was observed by Champenois : 2 " Fractions of fractions occur 
more rarely than the others (simple fractions), as two-thirds 
of three- fourths is written thus : f of f . Likewise, three- 
fifths of a half, f of i. Also, a half of two^thirds is i of f . 
Two-thirds of three- fourths of five-sevenths is f of f of f." 
Calandri, who placed multiplication of fractions before addi- 
tion, introduced the subject by finding the product of an in- 
teger and a fraction. 3 Though complicated by the use of 

1 Wencelaus, TFondament Van Arithnietica (1599 ed.). 

Rem eenen Silversmit iheeft een Item vn Orsebure a vn masse 

Masse Silvers van 15. Marc/ inde dargent de 15. marc. & ij. d'once, 

1. -|. once/ daer wt wil hy een il en vent faire vn ouurage de 8. 

weeck toerichten van 8. marc, ende marc. 5^. once, la demande est, 

5. X. oncen. De vraghe is : Hoe marc. 5^. once, la demande est, 

veel Silvers rester nodi? P. 82. 82. 

2 Champenois, Les Institutions De L'Arithmetique (1578 ed.). " Les 
Fractions de Fractions aduinnet plus .rarement que les autres, & f efcriuent 
par plufieurs fimples minutes, comme deux tiers de trois quarts ainfi fig- 
ure, -| de J. Item trois einquiefmes d'vn demy, | de -J. Plus vn demy de 
deux tiers -J de §. Dauantage deux tiers de trois quarts de cinq feptiemes 
§ de J de f " P. 88, fol. Giiij verso. 

3 Calandri, Arithnietica (1491 ed.). The problem is: "A man gained 
45 <y 15 I 8 4 6 in one year, what will he gain in 23 years 3J months?'' 

Original. "Lhuomo guadagna lanno 45 ^ 15 § 4 ( j che ghuadagnera 
in 23 anni 3 mefi £." (See next page for the solution.) 



THE ESSENTIAL FEATURES 



105 



denominate numbers it amounts to finding"^ of 45, 15, and 4. 
This is the appropriate phase with which to begin the multipli- 
cation of fractions and, as such, should precede formal work 
in addition and subtraction. Such a plan would be an im- 
provement in modern arithmetic. The directions for forming 
the products of two fractions were the same as those in present 
use : Take the product of the numerators for a new numerator 
and the product of the denominators for a new denomina- 
tor. 1 Thus, f met f comt -fy. Mixed numbers were usu- 
ally reduced to improper fractions, as in : 2 

met 

¥ ¥ 

A rare representation of the product of three fractions was 
given by Champenois : 3 I wish to reduce a half of two-thirds 



2 3 3# 



He solves the problem thus : 

AC T C 






45 


— J-0 


4 






3 


16 


-3* 







15 


- 3 




920 


1 


12 




115 


1 


5 




17 — 


- 5 — 


— 




— 


- 7 — 


— 8 




9 — 


— — 


— 


< 


2 — 


- 8 — 


— 




— 


— — 


— 9 




— 


— — 


— 1 




2 — 


- 5 — 


— 




— 


— — 


— 9 



1066 «y< 7/3 zs 

He will gain in the time stated above <y 1066 P 7 d 3- 
"Guadagnera nel fopira decto tempo difopra ^ 1066 /3 > $ 3 a pl'ii." 
Fol. dvi recto, fol. 26 r. Another example from Calandri is: "El cogno 
de uino vale 37 <y 15 /5 8 6 che uarranno 1 g cogna 5 barili ^3. Fol. d vl 
recto. 

1 Van der Scheure, Arithmetica (1600 ed.), fol. E 7 recto. 

2 Van der Scheure, Arithmetica (1600 ed.), fol. E ? verso. 

8 Champenois, Les Institutions De L'Arithmetique (1578 ed.). "Ie veux 
£. Ie multiplie le premier Numerateur 1. par le secod Numerateur 2. Ie 
reduire vn demy de deux tiers de fix feptiefmes ainfi figure -J de § de 
.pduict done 2. que ie multiplie par le troifiefme Numerateur 6. le produict 



io6 



SIXTEENTH CENTURY ARITHMETIC 



of six-sevenths, represented in figures thus: i of f of f. I 
multiply 'the first numerator, i, by the second numerator, 2, the 
product is 2, which I multiply by the third numerator, 6. This 
gives the product, 12, for the numerator of the reduction. 

" I then multiply the first denominator, 2, by. the second, 3, 
and obtain 6 as the product, which I multiply by the third 
denominator, 7, giving a product 42 for the denominator of the 
reduction. Then -J of f of f is reduced to one fraction, 
making Jf, or f." 

The meaning of the above multiplication was shown gra- 
phically by the following diagram : * 



p 




Q 












M 




N 


O 




m 




n 







i k 


' 




a 


c 


d 


e 


f 


g 


h 




a 


c 


d 


e 


f 


g 


h 


b 



The base line, ah, represents a unit and ah is f of it. Then 
af is f of f, and ad is \ of f of f . 

Trenchant 2 gave a diagram, which actually demonstrated 

dbnne 12. pour le Numerateur de la reduction. Apres ie multiplie le pre- 
mier Denominateur 2. par le fecond 3. le produict donne 6. que ie multiplie 
par le troif ief me Denominateur 7. le produict done 42. pour le Denominateur 
de la reduction. Parquoy -1 de J de f font reduicts en Vne Fractio fgauoir 
-ij-iemes, ou |ieme." Page 89, fol. Gv recto. 

1 Champenois, Les Institvtions De L'Arithmetique (1578 ed.), p. go, fol. 
Gv verso. 

2 Trenchant, L'Arithmetique (1578 ed.). 

MULTIPLICATION OF FRACTIONS. 

" Pour en auoir demonftration plus ruffirante : veyez 
cete fuperfioe ou quarre A, B, qui eft le produit de la 
ligne A, C : multiplie en Toy meme, laquelle faut entendre 
vn entier diuife en 5. Or eft il certain que 1 qu'elle rep- 
refente multiplie en foy fet 1, denotant le total quarre A, 
B : mais le 1 d'icelle, f cauoir eft, D, C, en foy multiplee, 



"3i 



1 ~~ 



THE ESSENTIAL FEATURES 



107 



the truth of the multiplication, a thing unique among the treat- 
ments examined. Thus he demonstrated that J X I = A, 
1X1= A* A X A = A> and so on. It is evident that 
the products of other fractions may he found from similar dia- 
grams. The applications of multiplication involving denomin- 
ate numbers were of the type, — the product of an integer and 
a fraction, — as illustrated by this problem from Kobel. 1 " 76 
persons have 4698 gulden to be divided among them. The 
question is : ' How much belongs to each ?' Solving by the 
Rule of Three, one finds that the share of each is 71 if 
guldens. Reduce further this fraction and all resulting frac- 
tions according to the method described in the rule above and 
according to the example which follows, until the lowest de- 
nomination and the lowest fraction of that denomination is 
reached." 

Division 
The prevalent order in the treatment of the division of frac- 
tions was ( 1 ) to divide the numerators of fractions having a 
common denominator, (2) to reduce to a common denomin- 
ator and divide the numerators, (3) to multiply crosswise the 
numerator of each fraction by the denominator of the other, 
(4) to invert the divisor and multiply. The third method is 
not so frequently used as the first and second, and the fourth 
is very rare. An excellent example of the fourth method is 
From Thierf eldern : 2 " When the denominators are different, 

ne fet que le quarreau E: qui n'eft que le JL- du total & enitier quarre 
A, B, qui en contient 25 femblables. Parquoy apert que -J- par J rnulti- 
plie, ne fet que -^ d'entier : comme auf f i | par 1 ne fet que -^ ; & § par 

" Semblablement f e peut demontrer par vne autre fuperfice, ayant 4 -de 
long, & 3 de large : que 1 par J, fet ^ ; & f par |, fet -^ : & ainfi des 
autres, ce qui nous a femble toon de declarer en paffant." Fol. H 3 recto. 

1 Kobel, Zwey rechenbuchlin (1537 ed.). "Auff das nim dii3 Exempel. 
76. perfonen haben vnder fich zu teylen 4698. gulden. 1st die frag/ 
was vh wie vil gebiirt jr jeglichem. Machs nach der Regel de Tri/ To 
findet fich das einem gebiirt 71. guide/ und i|- eines guldens. Dife vnd 
all ander briich reducier/ minder/ und bring fie alfo nach aufi weifung 
obgefchribner regel vnd Exempel/ wie naohuolgt/ in den kleyneren bruch 
oder in das kleyner theyl." Fol. H 8 verso. 

2 Thierfeldern, Arithmetica Oder Rechenbuch Auff den Linien vnd Ziffer/ 



108 SIXTEENTH CENTURY ARITHMETIC 

invert the divisor (which you are to place at the right) and 
multiply the numbers above (new numerators) together and 
the numbers below (new denominators) together, then you 
have the correct result. As, to divide | by f, invert thus 
i X! = H = iJ." Thierfeldern used cancellation to 

simplify the work, as shown in the ex- 1 3 

ample 1 at the right. This was rarely ~ mit -y?facit-| 

done in the works of that time. (4) (2) 

Teachers often complain that their pupils do not readily pass 
to the multiplication by the reciprocal of the divisor after they 
have begun by using the common denominator method. The 
crosswise method of the sixteenth century forms a connecting 
link between these two plans, and it is possible that it could 
be used to advantage in making the transition in teaching. 
The relation is shown in this example from Baker. 2 The 
divisor was generally written first (which would 9 

not be done now) and the terms of the result above f X f 
and below. If these fractions were changed to 
fractions with a common denominator, 12, the numerator 
would be found by multiplying the same numbers that are 
multiplied in the above work; and, since the result is the quo- 
tient of these numerators, Baker's process is the same as the 
one in which the common denominator is used, only the 
changed fractions are not in evidence. 

The denominate number problems following division were 
of this type : 3 "At Breslau a man buys 3 sacks of wool weigh- 
ing 14 stein, 12 stein, and 15 stein. How many centners is 
this, if one centner is equal to 5^ stein, and a stein is equal 
to 24 tb. ? Ans. 7 eeh 2. stein 2 lb. 

(1578 ed.). "Da aber die Nenner vngleich/ fo kehre allzeit den Theyler 
(welchen du zur rechten Hand fetzen folt) vmb/' vnd mukiplieir darnach 
die obern vnd vndern mit einander/ fo halt du es verricht/ Als : | in -|. 
stehet vmb gekehrt also : 
?4 



?0 



ii das Facit." Pages 64 and 65. 



1 Thierfeldern, Arithmetica (1578 ed.), page 63. 

2 Baker, The Well Spring of Sciences (1580 ed.), fol. Kviii recto. 

3 Rudolff, Kunstliche rechnung mit der Ziffern und mit den Zalpfennige/ 
(1534 ed.). 



THE ESSENTIAL FEATURES IO g 

" 12 Nuremberg pfennings equal what part of a pound? 
Ans. f." 

" I H I 6 is what part of I L? Ans. Air £" 
Doubling and halving were often included under fractions 
with the same meaning as under integers and with as little 
use for independent existence. Baker, whose work is char- 
acterized by minute classification, devotes the eighth chapter 
in his book to duplation (doubling), triplation, and quadrupla- 
tion of fractions. He gives these examples : 1 

(i) Duplation. To double any fraction, divide by i 

6 
To double f . J X §. 

8 
(2) Triplation. To triple f , divide f by -1 = |. 

Proofs for the operations with fractions were far less nu- 
merous than for those with integers. One would expect this 
to be so. The work with fractions, having to do with small 
numbers, had usually been done mentally, without the use of 
the abacus, and so without proofs. Hence there were no 
proofs to carry over into figure reckoning, as there were in the 
case of integers. Only one-eighth of the writers gave proofs 
in fractions and commonly placed them at the close of the 
treatment of each operation, or in a list at the end of the 
chapter. 

This table from Van der Scheure shows that each operation 
may be proved by applying the inverse operation. 2 
f Additio "| Soubstractio 



De proeve J Substractio I Additio 



van 



Multiplicatio 
Divisio 



Divisio 
Multiplicatio 



Commercial arithmetics, especially those which did not have 
denominate numbers under each operation with fractions, con- 
tained a section following the operations called the " Rule of 
Three with Fractions." Van der Scheure opens this section 

1 Baker, The Well Spring of Sciences (1580 ed.), fol. Liii verso, Liiii 
verso. 

2 Van der Scheure, Arithmetica (1600 ed-.), fol. E recto. 



HO SIXTEENTH CENTURY ARITHMETIC 

containing forty-one problems thus : x Having learned to use 
the species understanding^, do not fail to heed the advice to 
turn your thoughts quickly to the Rule of Three with Fractions. 

The following serve to illustrate the application of this 
rule : 2 

" If a centner of anything costs 9% florins, how much does 
a pound cost ?" 

"If 45 ells of cloth cost 13 ft 17 gr., how much do 7 ells 
cost?" Ans. 2 H 3 gr. 1 6 o Yz hi'. 

" 4^ braza are worth 17 sol., what will 8 braza be worth? 3 

" If a centner of wax is worth 13% florins, how much are 
17^ tb. worth?" 

PROGRESSIONS 

Among those arithmeticians on whom tradition worked its 
spell, none failed to give progressions (Arithmetic, Geometric 
and often Harmonic) a place. This subject appealed to both 
the classical and the modern scholars of that time, since it was 
persistent in the few classical works that survived and in the 
more recent acquisitions from the Hindus. Theoretical writ- 
ers did not attempt to justify the presence of Progressions in 
their text-books. That they were in existence was deemed 
sufficient reason why all educated persons should know of 
them. 

Although their position in the list of species was variable, 
the favorite place for them was after the division of whole 

1 Van <ier Scheure, Arithmetica (1600 ed.). 

"Die Specien gheleert. 
Zijn nil met claer bedien, 
Das laet den moet niet vallen, 
Maer snel u sinnen keert 
Toto den Reghel van Drien, 
Ghebroken in ghetallen." Fol. E g verso. 

2 Riese, Rechnung auff der Linien und Federn/ (1571 ed.). 

" Item/ Ein Centener fur 9 floren/ vnd 1 orth/ Wie kompt ein pf und ?" 
Fol. Dv recto. 

"Item/ 45 Elln tuchs fur 13 #/ 17 gr/ Wie kommen 7 elln? Facit 2 A/ 
3 gr/ 1 5/ o hi'/ y 3 r Fbl. Dvi recto. 

3 Borgi, Arithmetica (1540 ed.). 

C E fel te fuffe detto fe braza .4^2. de tela val. fol. 17. die valera braza 

.8. Fol. E K recto. 
5 



THE ESSENTIAL FEA TURES 1 1 j 

numbers. 1 In some cases they followed proportion, in others 
roots. 2 Kobel and Van der Scheure reserved the subject for 
the latter part of their works, after practical arithmetic had 
been completed, and Noviomagus deferred it to his second 
book, entitled Liber Secundus Arithmeticae, qui est de numer- 
orum Theorematis. 3 

The usual treatment consists in denning the series and in 
giving the rules for the last term and for the sum of a specific 
series. No proofs for these rules were attempted in arithmetic. 
This treatment of arithmetical progression from Tonstall 4 is 
more elaborate than those commonly given : 

"Arithmetic progression is the collection, into one whole, of 
1 Riese, Baker, Tonstall, and Buteo. 2 Finaeus. 

3 Noviomagus, De Numeris Libri II. 

4 Tonstall, De Arte Supputandi (1522 ed.). " Progressio Arithmetica 
eft numeroru inter fe ?qualiter diftantium in unam fummara eollectio: 

Eius autem duae funt fpecies. Altera eft: in qua naturali numer- 

orum ferie feruata, numerus quilibet fequens fola unitate praecedentem 
fuperat: ficut in hoc exemplo. 1.2.3.4.5. 6.7.8.9. Altera in qua numeros 
quoflibet omittentes, et paria feruantes interualla, longam numerorum 
feriem connectimus. velutei. 1.3.5.7.9.11.13. ." Fol. M x verso. 

" Ita fiet : vt numerus ex hoc productus lummarn omniu commonftret. 
veluti in hoc exemplo .1.2.3.4.5.6.7.8. primus numerus .1. ad poftremum .8. 
addatur et fient .9. Cumqj in tota ferie fint .8. loca. ducamus .9. in 4, 

eorum -dimidiu. et prodibunt .^6. quae omnium eft fumma. ." Fol. M 2 

recto. 

"Q7 Ti numeroru a fe equaliter diftantium atq} ordine continuo difpofi- 
tonim feries erit impar: tunc numerus indicans, quot loca funt in ferie, 
non in eum numerum ducatur : qui indicat quotus locus eft in ferie medius, 
fed in eum numerum qui in ferie medius reperitur : atqj ab utroqj extremo 
aequaliter diftaL Ita numerus procreatus omnium fumma patefaciet. ficuti 
in hoc exemplo .1.2.3.4.5.6.7. quia loca feriei funt .7. et medius numerus eft 
.4.7. in .4. ducamus : eit fient .28. quae fuma eft uniuerforum. Itidem I i 
exempli caufa fumantur. 1.4.7.10.13. quia loci ferie funt .5. et medius numer' 
eft .7. 5. in .7. ducamus: et fient 35. quae fumma eft omnium." Fol. M 2 
recto. 

"In omni progreffione Arithmetica, fiue feries par., fiue impar fuerit: 
numerus ab extremorum additione collectus in numeru indicatem, quot loca 
funt in ferie, multiplicetur. numerusqj productus postea dimidietur. et 
fumma progreffionis liabebitur. Exemplum in ferie pari. 1.3.5.7.9.11. pri- 
mus numerus additus ad postremu facit .12. et quia .6. loca feriei funt .12. 
per .6. multiplicemus. et f urgent .72. quae fi dimidientur: fient ^6. quae 
fumma eft progreffionis." Fol. M 2 verso. 



112 SIXTEENTH CENTURY ARITHMETIC 

numbers equally distant from one another. There are two 
kinds, one is the natural number series, i, 2, 3, 4, 5, 6, 7, 
where the numbers differ only by 1 . The other in which any 
number of terms of natural series is regularly omitted, as 
J > 3> 5> 7> 9> ri > !3- The sunn of the series 1, 2, 3, 4, 5, 6, 7, 8 
is found by adding 1 and 8, the first and last terms, and mul- 
tiplying this result by half the number of terms, f , or 4. It 
gives 36, which is the sum of the series. Similarly in 1, 3, 5,. 
7, 9, 11; (1 + 11) f = 12-3 = 36. In the case of an odd. 
number in the series, the sum of the series is equal to the 
middle term of the series multiplied by the number of terms. 
Thus, in 1, 2, 3, 4, 5, 6, 7; 7-4 = 28, the sum, of the series : 
and in 1, 4, 7, 10, 13, there are 5 terms and 7 is the middle 
one. Then 7-5 = 35, the sumi of the series. 

" In sail cases of arithmetic progression, whether the nun>- 
ber of terms be odd or even, the sum of the series may be 
found by adding the first and last terms, multiplying the result 
by the number of terms and dividing this result by 2. Thus,, 
*> 3. 5»7» 9> IJ ; C 11 + 1) 6= 126 = 72. 72 -±-2 = 36, 
the sum of the series." 

The following treatment of geometric progression from 
Adam Riese is a typical one from commercial arithmetic : 1 

" When numbers follow each other in twofold, threefold, 
or fourfold ratio, and so on, and you wish to find the sum, 
multiply the last number by the rate of progression, subtract,, 
the first term and divide this result by the rate of progression 
minus 1. 

2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048. 

2048 X 2 = 4096 4096 — 2 = 4094. 

-fAM_ = 4 o 94 , sum." 

1 Riese, Redlining auff der Linien und Federn/ (1571 ed>-). 

"So aber eine zal die ander vbertrit/ zweyfeltig/ 'dreyfeltig/ vierfeltig/ 
t-c. vnd wolteft die Summa wirien/ £0 multiplicire die letzte zal mit der 
vbertrettung/ nim von folchen die erfte/ was bleibt/ theil ab mit der vber 
trelttung/ weniger 1/ Als hie in folgenden Exempeln. 

"Item/ 2.4.8.16.32.64.128.256.512.1024.2048. duplir 2048/ komen 4096/ nim 
ab 2/ bleiben 4094/ die teil ab mit 2/ weniger 1/ als/ 1/ bleibt die zal an. 
jr felbs." Foil. €ij recto. 






THE ESSENTIAL FEATURES 



113 



Thierfeldern combines geometric progression with the ex- 
traction of roots. 1 " Hiernacht folget die Tafel/ dadurch 
ausgezogen werden alle Wurtzeln Geometrischer Progres." 



tf 


X 










} 


2 


i 






cr 


3 


3 


i 




M 


4 


6 


4 


i 






(3 


5 


10 


10 


6 


i 






n 


6 


16 


20 


16 


6 


i 






i&fi 


7 


21 


36 


36 


21 


7 


X 






w 


8 


28 


66 


70 


56 


28 


8 


¥ 




ccj 


9 


36 


84 


126 


126 


84 


36 


9 


y 



The characters in the left hand column represent the succes- 
sive powers of a number from the first to the ninth; that is,, 
these were algebraic symbols for x, x 2 , and so on. The vari- 
ous lines are seen to be the binomial coefficients, and the form 
is essentially that of Pascal's triangle. 

The traditional applications of the progressions were com- 
monly given, but when an attempt was made to supply vital 
applications, the results were curious. The following are 
from Baker : 2 

"A marchant hath sold 100 kerfies after this manner fol- 
lowing, that is to fay, the first peece for 1 s, the second peece 
for 2 s, the thirde for 3 s, and so foorth, rifing 1 s in every 
peece of kerfey unto the 100 peeces. The queftion is to know 
how much he fhall receiue for the fayd 100 peeces of kerfeys." 
This is evidently an attempt at giving a practical problem. 

" I woulde laye 100 f tones or other thinges in a right line, 
of every of the fayde f tones to be a juft pace one from an 
other, and one pace of from the fir ft ftone, there ftandeth a 
bafket. I demaunde howe manye paces a man fhall goe in 
gathering up the fayde f tones, and bearing them unto the 

1 Thierfeldern, Arithmetica Oder Rechenbuch Auff den Linien vnd Zif- 
fern/ (1587 ed.), page 331. 

2 Baker, The Well Spring of Sciences (1580 ed.). Fol. Fiii recto. 



ii4 



SIXTEENTH CENTURY ARITHMETIC 



bafket, the ftone after the other." This was a puzzle prob- 
lem of long standing. 

" There is one man departeth from London to Chefter, and 
fo to Carnaruan, the diftaunce beeing about 200 myles: He 
goeth the fyrfte day 1 mile, the fecond day 2 myles, the thirde 
day 3 : and so orderlye by natural progreffion. An other man 
departeth at the fame inftante from Carnaruan to London, and 
goeth the fyrfte day 2 myles : the fecond day 4 myles, the 
thyrde day 6 miles, and so encreafing every day 2 miles. 
The queftyon is to know in howe manye dayes they two per- 
sons shall meete togither." This is a courier problem. 

The following, though an artificial business problem, seems 
more plausible: "A man oweth me 400 l'i, to be payd in 10 
yeares, by progref fyon Arithmeticall, that is to fay 40 l'i at the 
end of the fyrfte yeare, and euery yeare following 40 l'i, to the 
end of 10 yeares, hee offereth to pay me the fayd 400 pound 
al at one paiment. The question is to know at what time hee 
ought to paye mee the fame at one paimente, that I be not 
intereffed in the time." 

Only one problem in geometric progression was given. It 
is this : 

"A marchante hath folde 15 yeardes of Satten, the firfte 
yarde for 1 s, the fecond 2 s, the thyrde 4 s, the fourth 8 s, 
and so increafing by double progreffion Geometricall. The 
queftyon is to know, how muche the fayde marchant fhal re- 
ceiue for y fayd 15 yardes of fatten." 

Although the French writers were chiefly theorists, Cham- 
penois wrote to meet business needs, and his introduction to 
this chapter reads r 1 " Progression is a series of numbers in 
which there is a certain excess between the numbers, and the 
first is always exceeded by the second as much as the second 

1 Ghampenois, Les Institvtions De L'Arithmetique (1578 ed.). 

" Progref f ion eft vne fuite de nombre, qui out vn eertain excez les vns 
entre les auitres, & toufiours le premier eft autant excede du fecond, que 
le fecond du troifiefme, & iainfi des autres. L'vfage de la Progreffion eft 
vai compendium d'Addition, & eft fort vtile, tant en diuerfes questios 
d'Arithmetique, Geometrie, Mufique, qu' Aftrologie, la ou plufieurs regies 
font faiotes par la nature de la Progreffion," Fol. Eviij verso. 



THE ESSENTIAL FEA TURES 1 1 5 

is by the third, and so on with the others. The practice of 
progression is a summary of addition and is very useful in 
questions of arithmetic, geometry, music, and astrology, where 
many rules are made in the nature of progression." 

Among his problems in progression are the following : 

"A merchant has 60 horses and sells them to another mer- 
chant; he is to have 4 ecus for the first, 8 for the second, 12 for 
the third, and so on for the others. The question is : ' How 
much should he receive for the last, and how much should 
he receive for all ?' " x 

"A man about to' die gives all his money to his relatives on 
the condition that the first should have 4 ecus, the second 6 
more than the first, and the amount for the third and the 
others should increase at the same rate to the last. After 
the distribution was made, it was found that the last had 
118 ecus for his part. The question is: 'How many rela- 
tives had he and how many ecus ?' " 2 

"A merchant sells a piece of velvet containing 16 aunes to 
another merchant for which he shall give 2 sous for the first 
aune, 6 sous for the second, and for the others he shall give 
at the same rate up to the sixteenth aune. The question is : 
' How much should he give for the 16 aunes?' " 3 

"A gentleman has 12 horses, which he wishes to> sell, and 

1 " Vn marchant a 60 cheuailx, & les vend a vn autre niarchant, par tel 
fi qu'il doit auoir 4 efcus du premier, 8 du fecond, 12 du troifiefme, & 
ainfi des autres. On demande combien il doit receuoir du dernier, & com- 
bien il doit receuoir du tout." Page 63, fol. Eviii recto. 

2 " Vn homme allant de vie a 'trefpas, donne tous fes efcus a f es> parents, 
par telle codkion que le premier doit aubir 4 efcus : le fecond 6. plus que 
le premier, & ainfi du troifiefme, & autres iusques au dernier. Et apres le 
paitage faict Ion a trouue que le dernier ia eu 118. efcus pour fa part. Lon 
demande combien il auoit de parents, & combien d'efcus." Page 64, Eviij 
verso. 

8 " Vn marchant vend vne piece de velours, qui contient 16. aulnes, a vn 
autre marchant, par tel fi qu'il donnera 2. fols pour la premiere aulne, & 
6. fols de la feconde, & ainfi pourfuiuant la Progref fio des autres iufques 
a la feiziefme aulne. Lon demande combien il doit des 16. aulnes." Page 
66, fol. F verso. 



n6 SIXTEENTH CENTURY ARITHMETIC 

gives the first for 3 sous, the second for 12, the third for 48 r 
and so on with the others up to the last." * 

"A dealer wishes to sell a robe to a Master of Arts, who 
has little money. The Master of Arts, however, seeing* him- 
self without a robe, and that this one for which he was bar- 
gaining, seems to be well made, says to the dealer that he will 
give 4888 livres for it, reduced twenty times by half. The 
dealer is well satisfied and gives the robe to the Master of 
Arts, who does not refuse it, but joyfully putting it on his 
shoulders makes his reckoning with the dealer. At the end 
of the calculation he finds due to the dealer 2 T f^J^ deniers, 
which is worth f of a quarter of a pite, and -ffths of ye of 
a pite. To pay this large sum he takes from his purse a piece 
worth 3 deniers, gives it to the dealer, and demands his 
change. The dealer driven to despair asks if he will put it at 
stake. The Master of Arts agrees. The dealer loses, the 
Master of Arts gains, and taking his 3 deniers he joyfully 
takes leave of the dealer. The dealer thanks him, saying that 
he is at his service. Thus, the Master finds himself well 
dressed at little expense by means of the cunning of an arith- 
metical rule." 2 

1 " Vn Seigneur a 12 cheuaux, qu'il veut vedre, & donne le premier pour 
3. fols, le fecod pour 12. le troifieffme pour 48. & ainfi des autres iufques 
au dernier." Page 67, Fii recto. 

2 Champenois, Les Institutions (1578 ed.). "Vn frippier veut vendre 
vne robbe 58. lines a vn maiftre es arts, qui n'auoit pas beaucoup de pecune.. 
Toutefois fe voyant fans robbe, & que cell© qu'il marchadoit, luy fembloit 
bien faicte, diet au frippier qu'il en doneroit 4888 liures, en rabatamt de 
rooitie iufques a 20 fois. Le frippier fut contenlt, & dona la robbe au 
maiftre es arts : ilequel ne la refuf a pas, mais id 1 ' vne gayete de coeur la mit 
deffus fes efpaules, & fit copte auec le frippier. Et a la fin du copte fe 
trouua redeuable au frippier de 2. deniers & _2 i_7._,i eme .d'vn denier qui 
vallet ^4 'd'vn quart de pite, & -|Jieme d'vn quart de quart de pite. Bt pour 
faire le payement de cefte grande JOomme, preden fa bourfe vne grande 
piece de trois deniersi, & la done au frippier, & demande fon refte. Le 
frippier tout defefpere luy demanda f'il vouloit iouer fon refte a fa grande 
piece de trois deniers. Le 'maiftre es arts diet qu'il eftoit cotet. Le frip- 
pier perd: le maiftre es arts gaigne, & pred fa piece de trois deniers, & 
doyeufemet prend conge du frippier. Et le frippier le remercia, & luy diet 
que luy & fon bie eftoiet a fon comiandement. Monfieur le Maiftre se 
trouua bien pare a peu de frais, par le moyen de la fubtilite d'vne regie 
d'Arithmetique." P. 70, fol. Fiij verso. 



THE ESSENTIAL FEATURES 



117 



RATIO AND PROPORTION 

The treatment of this subject, more than that of progres- 
sions, followed the traditional lines of Greek and Hindu 
works * and occurred less often in the practical arithmetics of 
the period. The most useful tool in the solution of commer- 
cial problems, the Rule of Three, was, of course, proportion, 
but it was seldom connected with that subject. 

It was the universal custom borrowed from the Greeks to 
use the terms, proportion and proportionality, for ratio and 
proportion until the latter part of the sixteenth century. Ton- 
stall stated the Greek meaning of proportion in such a way as 
to enable one to recognize at once the corresponding modern 
ideas of Ratio and Proportion and gave an explanation that is 
more than a mere category. His treatment was as follows : 2 

" Proportion is nothing else than a comparison of things 

1 Unicorn, De L/Arithmetica vinversale (1598 ed.). " Che contien li aftri 
4. Algorismi della prattica di Arithmetics, cioe de radici, del piu, & men, 
de binomii, & recisi, & de proportioni, -de radice uniuersali, & estrattioni 
de radici delli binomii, & recisi, quali sono il neruo del decimo libro de 
Euclide, & della proportione hauente il mezzo, & doi estremi, della qual il 
decimoterzo libro di Euclide." Fol. Fi recto. (Besides the four algorisms 
of practical arithmetic, this contains roots, positive and negative number, 
binomials, surds, proportion, roots in general, the extraction of roots of 
binomials and surds, which are contained in the tenth book of Euclid, and 
the extremes and means of proportion from the thirteenth book of Euclid.) 

2 Tonstall, De Arte Supputandi (1522 ed.). "Ilia itaq* habitudo, qua 
f ef e uel aequaliter, quando f unt fsquales : mutuo refpiciunt. uel in^qualiter, 
quando earum altera maior reliqua, aut minor eft: appellator proportio. que. 
nihil aliud eft : q earum inter f e comparatio eiuf dem quoq^ generis quan- 
titates effe 'debent: inter quas cadit proportio. Veluti duo numeri, duae 
lineae, due fuperficies, duo corpora duo loca, duo tempora. neq5enim linea 
maior aut minor fuperficie eft, aut corpora: nee tempus.loco maius eft, aut 
minus, fed linea linea: fuperficies fuperficie: corpus corpore. fola enim, 
•quae unius funt generis : inter fe comparabilia funt." Fol. i t verso. 

" Quippe proportio apud ueteres in tria fecatur genera, quorum unum eft 
difcretorum, uidelicet numerorum: quod uocant Arithmeticum. Alterum 
continuorum : quod geometricum appellant. Tertium fonorum et concen- 
tuum : quod armonicu nuncupant, ex illoru utroq3 mixtu : q? mufica in 
paufis et prolationibus tepus fpectet : in uibrante nocu notarumq? diuifione, 
numeros." Fol. i\ verso. 

"Vnde fit: ut qu^cumq? proportio occurrit in numeris : eadem reperiatur 
an omrii genere continuorum, puta in lineis, fuperficiebus, corporibus, et 
temporibus." Fol. i 2 recto. 



n8 SIXTEENTH CENTURY ARITHMETIC 






among themselves. Quantities which formi a proportion must 
be of the same kind, as two numbers, two lines, two surfaces, 
two solids, two like orders, and not, a line is greater than a 
surface, or solid, or time is greater or less than a space, but 
line is comparable with line, surface with surface, solid with 
solid, and so on. 

"Among the ancients proportion is divided into three 
classes, arithmetic, geometric and harmonic. 

" Just as proportion occurs in numbers, so it is found in 
all continua, as in lines, surfaces, solids, and time." 

Kinds of Rational Proportion 1 
"Any quantity compared with another is equal or unequal to 
it. A quantity is equal to another when it does not exceed 
that quantity, or is not exceeded by it, as a cubit to a cubit, or 
a foot to a foot, and so on. Unequal quantities are those in 
Which one quantity exceeds the other. A series of numbers 
having a common difference such that the difference is an 
integral part of the first number is said to be composed of 
proportional numbers, and the series is classified according to 
the name of the part. They are called sequialteri, sequitertia, 
and so on. This is the origin of arithmetic progression." 

Tonstall here gave the Pythagorean table again, showing 
ratio, proportion, multiplication, and division in their inter- 
dependence. 2 " Proportionality is the similarity of propor- 
tions among themselves." In this chapter Tonstall gave 
arithmetic and geometric proportionality and referred to 

1 Tonstall, De Arte Supputandi (1522). 

" PROPORTIONIS RATIONALIS SPECIES." 

" Omnis quatitas ad alia coparata aut ei ^qualis : aut in^qualis reperitur. 
Quantitas §qualis eft : quae nee fibi comparatam excedit : nee ab ea exce- 
ditur. Veluti cubitus ad cubitum collatus. pes ad pedem. numerus quater- 
miarius iad quatemarium." Fol. i 2 verso. 

" In^qualis autem quantitas, quae fibi comparatam excedit : proportionem 
ad ilia habet infqualitatis maioris : ueluti cubitus ad pedem. numerus quater- 
niarius ajd binarium." Fol. i 2 verso. 

2 Tonstall, De Arte Supputandi (1522). " Proportionalitas eft propor- 
tionum inter fe fimilitudo." Fol. K 4 recto. 



THE ESSENTIAL FEATURES 



IIQ 



Jordanus as having- given eight kinds. Vicentino gave four : 
arithmetic, geometric, harmonic, and contraharmonic. 1 This 
subject was given a more rigorous treatment by algebraists. 2 
Ratio and proportion in the modern sense came into arith- 
metic about 1550. Orontius Finaeus under Ratio, Propor- 
tion, and Rules of Proportion gave this definition : 3 " Ratio 
(as we shall use it) is the method of comparing two num- 
bers of the same kind." The same conception of ratio will 
be recognized in this example from Buteo : 4 " When two 
numbers are proposed, as for instance, 12400 and 124, the 
ratio between them will be immediately shown by dividing 
the greater by the less; since it is 100 in this case, we shall 
call the ratio between the given numbers a hundredfold ratio." 

1 Vincentino, Proportione, et Proportionalita (1573 ed.). 
" Proportionalita, fi dice la ugualita delle proportioni. 
'La proportionalita, fi divide in Arithmetica, Geometrica, Harmonica, et 
Contraharmonica. 

2 2 
" La Arithmetica 5 7 9 



69 
La Geometrica 4 6 



2 1 
"La Harmonica 643 



"L'una ch'ha la differentia dello antecedente fopra il confeguente, alia 
differentia del confeguente fopra il terzo quanto, f altra la proportione dell' 
•antecedente al terzo quanto. 
1 2 

"La contraiharmonic 653 

"L'una che ha la differentia del confeguente fopra il terzo quanto, alia 
differentia dell' antecedente, fopra il confeguente, l'altra la proportione dell' 
antecedente, l'altra la proportione dell' antecedente al terzo quanto." Fol. 
G ± verso. 

2 Tartaglia, La Prima Parte Del General Trattato (1556), Book 7. 

3 Finaeus, De Arithmetica Practica (1555 ed.). "Ratio igitur Cut ad 
remipfam deueniamus) eft duarum quantitatum eiufdem fpeciei adinuicem 
comparataru habitudo." Fol. Q 1 recto. 

* Buteo, Logistica, Quae Arithmetica vulgo dicitur (1559 ed.). " Pro- 
pofitis duobis numeris, vt puta 12400, et 124, quaena fit inter eos ratio, 
partitione maioris in minorem, ipfum prouenies ftatim oftendet, quod cum 
fit in hoc loco 100, dicemus inter datos huiufmodi numeros rationem effe 
centuplam." Fol. f g recto. 



I2 o SIXTEENTH CENTURY ARITHMETIC 

Ramus defined ratio as division. 1 He then added the old 
conceptions, as : " The ratio of 3 to 2 is sequialter, because 3 
contains 2 one and a half times. The ratio of 5 to 3 is super 
biteria, because 5 contains 3 one and f times." He defined 
proportion as the equality of two ratios. 2 Thus, proportion 
came to replace the old term proportionality, and at the be- 
ginning of the seventeenth century, the title, Ratio and Pro- 
portion, had taken the place of Proportion and Pro- 
portionality. 3 

Ludolf van Ceulen (161 5) gave a chapter on proportion as 
applied directly to practical problems. He says that pro- 
portion is necessary in comparison of weights and measures/' 4 
as when one says that a pound is heavier than a mark in com- 
paring weight, or that the distance from Utrecht to Delft is 
farther than that to Leyden, in comparing length." 

Ratio of fractions 5 and the Rule of Three 6 of fractions are 

1 Ramus, Arithmeticae Libri Duo (1577 ed.). "Ratio est eomparatio, 
quoties .terminus in termino oontinetur." Page 41. 

2 Ramus, Arithmeticae Libri Duo (1586 ed.). " Proportion*) arithmetica 
lie est, geometrica fequitur in rationem aequalitate." 

3 Stevin, Les Oeuvres Mathematiques (1634 ed.). " Des definitions de 
la Raison et Proportion." " La proportion, pour en parler un peu en gen- 
eral, avant que parvenir au partieulier, eft la fimilitude de deux raifons 
egales. Railon eft comparaifon de deux termes d'une mefme efpece de 
quantite." Fol. Bi verso. 

4 Van Ceulen, De Arithmetische en Geometrische fondamenten (1615 
ed.). "Alsmen spreeckt een pondt is svvaerder als een marcq dan ver- 
ghelijcktmen ghewiohte, ofte van Delfft is Vtrecht verder als Leyden, hier 
vvert de lenghte vergheleecken." Fol. Biij verso-. 

5 Buteo, Logistica (1559 ed.). 
To find the ratio of J and £ : 

I X | J The ratio is 21 to 20, or sesquivigesima. (See pages 
21 20 I 33, 34 of this article.) 
"De particularum fragmentorumque rationibus, quomodo dignofcantur." 
u Non folum numeri conferuntur numeris, fed etiam particulis, atque frag- 
xnentis, & ipfae etiam inter fe particulae. Quarum rationem inuestigabis 
hoc modo. Sit propofitum dare rationem quam habet -| ad |-. Difpo- 
natur datae particulae atque multiplicetur in decufiim, 
fientque producta 21 & 20. Quae eft igitur ratio 21 ,ad 20, 21 20 

eadem eft -g. ad ^, hoc eft fefquiuigefima." Fol. f 6 verso. | X f 

6 Ramus, Arithmetice Libri Duo (1577 ed.), Liber II, Chap. VI. 



THE ESSENTIAL FEATURES 



121 



subjects sometimes met. The latter is generally characteristic 
of the commercial writers. 1 

INVOLUTION AND EVOLUTION 

The commercial arithmetics naturally contain little concern- 
ing- powers of numbers, and one might expect to find much less 
on evolution. But there was a large number of problems in 
mensuration, a subject of real interest at that time, which re- 
quired a knowledge of roots. Consequently, most works of 
any pretension gave square and cube root. Theorists like 
Tartaglia 2 and Unicorn 3 explained evolution of orders higher 
than the third. It was quite common to preface the work on 
roots by a table of powers. 4 

1 Thierfeldern, Arithmetica Oder Rechenbuch Auff den Linien vnd 
Ziffern/ (1587 ed.), Chap. IIII. 

Baker, The Well Spring of Sciences (1580 ed.)- "If one elle coft me 
17 s" what fhall 15 elles ]/& part cofte? which l /s is halfe a quarter of an 
elle." Fol. Qvii recto. 

Kobel, Zwey rechenbuchlin (1537 ed.). "So einer dich fragt/ wan einer 
9. elen vnnd %. Tuchs vmb 7. gulden kaufft bet/ was in 6. Elen koften." 

" If a person bought g%. ells of cloth for 7 gulden, what will 6 ells 
cost?" Fol. I 2 recto. 

2 Tartaglia, La Seconda Parte Del General Trattato (1556), Book II, 
fol. Cvi verso. 

3 Unicorn, De L'Arithmetica vniversale (1598), Book 2. 

4 Champenois, Les Institvtions De L'Arithmetique (1578). Champenois 

gives the squares of numbers from 1 9, inclusive, illustrating each by 

little squares beneath, thus : 



Racines 


1 


2 


3 


4 


• 


« 


i 


8 


9 


Simples 


1 


4 


• 


16 


2S 


36 


49 I 


64 


81 




• 










====== 









Page 258, fol. S verso. 

The subjects of powers and roots were more generally treated in 
algebra or in algebraic chapters of the arithmetical works. The following 



122 



SIXTEENTH CENTURY ARITHMETIC 



At that time roots were extracted by at least four methods,, 
each subject to some variation. 

1. Method of line reckoning". 

2. Geometric method. 

3. Scratch method. 

4. Downward method. 

Although the methods by the use of figures were not sim- 
ple, the method by counters was still more difficult, and was 
rapidly superseded by the former. Examples solved by line- 
reckoning were given by Kobel, Riese, and a few others. Geo- 
metric illustrations of square and cubic numbers were not un- 
common, but graphical methods of extracting roots were rare. 
Trenchant in the third book of his Arithmetic gave the geo- 
metric demonstration commonly 
found in American arithmetics 
of the nineteenth century. In 
this he explained the construc- 
tion of the diagram as here 
given and proved the equality of 
the rectangles, AF and FD, by 
reference to Euclid, Bk. II, Prop. 
4. He then said : " From the 
above it is easy to see clearly 
and to demonstrate the logic of 

is a table from Ghaligai, Practica D'Arithmetica (1552 ed.), explaining his 
power symbols. 



40 


c 


7 


280 




49 




F 




2209. 
1600. 




280 



n o Numero 

o cosa 

a Censo 

ca , cubo 

o cU a dio 

Q — „ Relato 

* QtUa cd di a 

Pronico 

ad*ia<&-v o di o dio — 

azeUca » ca di en — 

g^o g di a 

S3 Tromico 

CD Ma di d cd di a di odi— 

Qj DTomico 

E3iU a — & di J3 



DJ • 



m 







1 

3 

4 

8 

16 

32 

64 

128 

266 

612 

1024 

2048 

4096 

8192 

16884 

82768 



Fol. Kii recto. 

* Error in original. It should be turned horizontally, "di" indicates "to the power." 
Thus, D di D means 4, DD di □ means 8 to the second power, DD di DD means 8 to the 
third power, and so on. 



THE ESSENTIAL FEATURES 



123 



extracting square root : for let us take the same number, 2209 
(which is the square 00 the whole line AB), to find the square 
root, the extraction of which we shall demonstrate from the 
figure and maintain the logic. First having separated the 
number into two periods (according to Article 4 of this chap- 
ter), I seek, according to Article 5, the root of the first period, 
22 : this is 4, which (because there is another root figure, or 
because it is of the tens' period) must be 40 and denotes the 
longer segment of the line, i.e., AC, and consequently, each of 
the sides of the square, EF, and also the longer sides of the 
supplementary rectangles, AF and DF. From 22 I subtract 
the square of 4, which is 16, and because 4 denotes 40, then 
its square 16 denotes 1600. This is the square EF. Then 
the remainder from 22 is 6, or from 2209 will leave 609, 
Which is the area of the two rectangles, AF, DF, and of the 
small square, BF. But, if I divide the area of a rectangle by 
one of its sides, I obtain the other. That is, if I divide the 
product of two numbers by one of them, I obtain the other. 
So if I divide 280 (the product of 7 by 40) by 40, or 560 
(the product of 7 by 80) by 80, I shall get 7. Therefore, 
knowing that 609 is the area of the two rectangles, AF, DF, 
and of the small square BF, and that 40 is the side of one of 
these rectangles, I double 40 making 80, denoting the side of 
the rectangles joined. Then I divide 609 by 80, so that I can 
also take away the square of the quotient, and obtain 7, which 
denotes the shorter sides of the rectangles, AF, DF, and con- 
sequently the side of the small square BF, which is BC. Fin- 
ally, I multiply 80 by 7, and then 7 by 7, producing 560 and 
49, which make 609; then I subtract the 609, and the re- 
mainder is zero. Thus, the square root of 2209 is 47." 2 

1 It is well known that the Western Arabs used the sand-board in ex- 
tracting roots; hence it is reasonable to suppose that they used the scratch 
method, whence the Italians obtained it. Cantor, Bd. I, p. 767 (igoo ed.). 

2 Trenchant, L'Arithmetique (1578 ed.), Book 3. "Par ce que deffus 
fe pent clerement veoir, & demontrer la rail on des extractions quarees : 
pourquoy fere prendrons ce meme nombre 2209 (qui eft le quarre de la 
totale ligne A, B) pour en tirer la ratine quarree: a I'extr action da laquelle 
rnonftreron'S fur la figure la raifon pretendue. 

" Premierement ayant coupe iceluy nombre en deux lections (felon le 4 



124 



SIXTEENTH CENTURY ARITHMETIC 



The accompanying diagrams from Trenchant's pages, for 
extracting the cube root of 103,823, show that he used the 
modern block method. 




Fol. Q 2 verso. 

The scratch method was the common algorism for this pro- 
cess. The following examples from Tonstall will illustrate 
it : * " The extraction of the square root is nothing else than 
the finding of a number which multiplied by itself will pro- 
duce the proposed number, if it be a square, or the greatest 
square number contained in it, if it be not a square." 

" Example in square root : 

artic. de ce chap.). Ie oherehe, felon le 5 art. la racine de la premiere 
fedtiom 22: c'eft 4, lequel (a eaufe de la figure radicale future, ou qu'il eft 
de la fections des dizeines) vaut 40 denotat la maieure lection de la ligne 
A, C, & par confequent, ehacun des cotez du quarre E, F, & auffi les plus 
grans cotez des fupplemens A, F, & D, F. En apres de 22, ie leue le 
quarre de 4, c'eft 16, lequel comme 4 denote 40, auffi fon quarre 16 deno- 
tera 1600, c'eft le quarre E, F. Par ainfi de 22 refte 6 : ou de 2209, ref- 
tera 609 qui eft la fuperficie de 2 fuplemens A, F, & D, F, & du petit 
quarre B, F. Or qui diuife la fuperfice d'vn rectangle, par l'vn de fes 
cotez vient l'autre: ceft a dire, qui diuife le produit de deux nombres par 
l'vn d'iceux, vient l'autre : comme f i ie diuife 280 (prouenu de 7 foys 40) 
par 40, ou 560 (prouenu de foys 80) par 80 viendra 7. Parquoy fachant 
que 609 eft la fuperficie des deux fuplemens A, F, & D, F, & du petit 
quarre B, F, & que 40 eft le cote de l'vn d'iceux fuplemens, ie double 40 
fet 80, denotant le cote des deux fuplemens affemblez. Donques ie diuife 
609 par 80, de forte que i'en pu'if f e auffi leuer le quarre du quotient, vient 
7 qui denote les moindres cotez d'iceux fuplemens A, F, & D, F, & par 
confequent ceux du petit quarre B, F, dont la fection B, C, en eft l'vn. 
Finaiblement ie multiplie 80 par 7, & encores 7 par 7, prouient 560, & 49, 
qui font 609, que ie leue de 609, & n'y refte rien. Ainfi la racine quarree 
de 2209 eft 47." Fol. P 6 recto. 

1 Tonstall, De Arte Supputandi (1522), fol. N. recto et seq. 



THE ESSENTIAL FEATURES 



125 



" The number g-iven, 57836029, should 2 

be properly pointed off. Beginning at £ * 

the left, we should search for the first JJ m t Reliquum 
number which multiplied by itself will - 7 6 5 xiadix 
give the first number marked off at the WWW 

left, or, if it is not a perfect square, the W 

number which multiplied by itself will give the nearest square 
which it can contain. The number is 7, and its square is 49, 
which subtracted from 57 leaves 8, and so 7 is placed between 
the parallel lines first made, 8 is put above the first g 
point and crossed out. Then the 7 between the . . . 

parallel lines is doubled, which makes 14, of -= 

which 4 is placed below and at the right, and the 

10 left should be placed below the lines directly below the 8, 
which was the result of the former subtraction. Then again 
another number is to be found which, multiplied by 
14, will give the number nearest to 88 (the 8 left .4. . 
from subtracting and the next figure of the given 57836029 

number.) This figure is 6, which multiplied by -= 

14 gives 84; this taken from 88 leaves 4; then 
this number 6 multiplied by itself gives 36, which subtracted 
from 43 (the 4 left from the last subtraction and the 
next figure in the given number) leaves the num- 
ber 7, which is written above the next point in .£. . . 
the number given-. The 6 is placed between the 57836029 

parallel fines as the next figure of the root, and -^ 

so on." 

" If one wishes to find out 
Whether the number found is 
right, multiply the number found 
by itself and add the remainder, if 
there is one. If the work is cor- 
rect, this result will be the given 
number." x 

The example in cube root 
shown in the illustration is a re- 
production from Tonstall's arith- 
metic : 2 



EXEMPLA nunc afteram as : que ll'ng ula imm'ftftct. 
Acqjexducena'esquinquagics milli'es millcnis miDibus, 
quingetuies uitfcs ter milleni's mi1l;bus,quingentisoftos 
gintaduobus millibus,quadrmgenas fexaginta quattB 
or, radicera cubicam eruamus. 

# if 

6 j o 4 Radixcubi 

X * X # * P 

# ff » * * # ft 

STAT1MQVE poftqnumcrifuoordmcpetfefpri: 
duftxqi fubcus parallels : et millcnarionim fedes pu&i'j 
«unt f ignate : fub p oicr cm o ad [in i fcram mQlenario puns 
<lo notatonumeni! ali'quis primarius exqui'ratnr : quifo 
tnelinfc.etLterumuinumci-iproduifcutn molcplicatof, 



1 Ton stall, De Arte Supputandi (1522), fol. N 2 verso. 

2 Tonstall, De Arte Supputandi (1522), N 4 vexso. 



SIXTEENTH CENTURY ARITHMETIC 

A peculiar form in which the process is the same as the 
scratch method, only the numbers are not crossed out, is found 
in Cirvelo: x 



















o 






o 


12 





(a) 


ool 


(b) 


ool 


(c) 





oo | 


(A) 


03 


00 


(e) 


i 


28 







16 | 




25 1 




I 


1 44 




07 


24 1 




5 


47 


56 



4 I 



5 I 



1 22 



8 162 I 



2 I 43 I 94 I 



4 I Radix. 5 I Ra. *i | Ra. 
* The 2 is missing in the original. 



82 



14 



2 I 3 I 4 I Ra. 



Problem (a) is the process of finding the square root of 16, 
(b) of 25, (c) of 144, in this, however, the root is incom- 
plete, (d) of 6724, and (e) of 54, 74, 756. The powers are 
written above the double lines and the roots below them. 

A downward process resembling in many respects our pres- 
ent form is found in Widman : 2 

To extract the square root of 207936 : 



(a) 


(b) 


(c) 


^7936 


47936 


5436 


8 


8 


90 


4 


45 


45 



The figures in the top row are the successive remainders with 
all periods brought down at each step. The last row is the 
root, the numbers in the root being repeated each time. The 
middle row is the trial divisor, not always having the zero 
added. Step (b) is really step (a) with the next figure, 5, 
of the root written down. 

An interesting plan is given by Paciuolo 1 , which has the 
principle common to all the other methods, but which differs 
from them in arrangement of calculation : 3 

1 Cirvelo, Arithmetiee practice seu Algorismi (1513 ed.), fol. b t recto. 

2 Widman, Behend und hupsch Rechnung (1508 ed.), fol. di recto. 

3 Paciuolo, Suma de Arithmetica (1523 ed.). 

" De aproximatione .R. in furdis." 
"Apresfo per le. R. forde: cioe qlle che no fonno difcreta: qui fequente 
mettaro vna re*, p laquale femp tu piu chel copagno te porrai aproximare : 
vrnde a voleir trouare ditt§. R . f empre troua prima la f ua. R . derita de 
poto: cotmo fai laltre d fopra. E qfi tu hai itrouato la prima. R. fanne 
pua. e vedi quanto la pasfa el detto n°. Alora torrai quel piu: cioe la 
dria e ptirala per lo doppia de qfta prime. R. che te la data: e quello 
die virra de ditto ptimento cauaralo de ditta pma . R . el remanente f era la 



THE ESSENTIAL FEATURES 



127 



To find the square root of 6 : 

(1) !/6T- - 2 +~ A = *X. (2) T/6T. . 2%+-~^- = 2 2 V 

(3) T/6T- - 2/0 + 6 T9 6 ^" ^ %VeV, and so on. 

This agrees with the general formula : 

A — a 8 , A — a 2 ! 

a = — — — = a x , a H = a 2 , and so on. 

2a 2a 

It is easy to recognize in every method the plan of dividing 
the remainder by twice the part of the root found and adding 
the result to the root for the next figure. 

Square and cube roots of fractions were common. Wid- 
man gives these examples: Find the square root of ff and 
the cube root of -ffo." " Give a number, \ of \ of \ of 
t of which is its own quare root." Ans. AWA- 1 

APPLIED ARITHMETIC 

The authors of arithmetic of the sixteenth century may be 
classified into three groups : writers of theoretic arithmetic, 2 

.R. fcda. <ie ditto n . afai piu p fimana ohe la p a . poi p aproximarte piu: 
farai la pua ancora di qfta. e vederai quato e la fupchia ditto n°. E anche 
quel piu che te dara qfta feconda. R. ptiralo pure per lo dopio de esfa 
.R. 2*. che te la dato : e qllo auenimento caualo de ditta .2 a . R. el remanete 
fia. . 3*. piu pximana de ditto n°. Poi a mo ditto: farane proua. e 
vederai quanto la pasfi ditto n°. e pigliarai anche quella dria e ptira la per 
10 doppio pure di quefta. R. 3 a . che tal dria te dette: e lauenimento de lei: 
caua el rimanente fera. R. quarta piu pfimana de ditto n°. E cofi Temp re 
in infinitu andarai facedo : e guarda f empre de cauare li ditti auenimeti de 
le R. fchieitte: e no delle duplate verbi gra. R. p a . di 6. ene. 2^. E qfta 
pasfa de. %. pche. 2^. via. 2^. fa. 6^4- parti quel. 34. quale e la dria 
p lo doppio di la R. prima che la dato: cioe p .5. neue -^. qual caua de 
.2y 2 . die e la. R fohietta resta .2 9 e quelto di co che. 2 a . R. di 6 piu 
pximana che la prima, cioe. 2^. E cofi andarai fequitando a modo ditto 
e trouerai che qlta 2 a . R. pasfa .6. de .4^-. la terza R fera .2 T 8_y_. 
E quefta pafsara .6. de - g g- g . i 60() . la quarta .R. fera fecondo che vedi q 
de fcripto in margine. e cofi la quinta al medifimo modo trouato. E 
quefito fin qua e detto f e habia intendere de li numeri f ani. Fol. 45 recto, 
F v recto. 

x Widman, Behend und hiipsch Rechnung (1508 ed.). " Gyb ein zal 
welch mit irm \, \, \, \ sey ir selbst qdrata radix. Ans. /oWsV Fol. E 7 
recto. 

2 De Muris (1538), Maurolycus (1575). 



128 SIXTEENTH CENTURY ARITHMETIC 

writers of arithmetic primarily theoretic and secondarily prac- 
tical, 1 and writers of arithmetic primarily practical and sec- 
ondarily theoretic. 2 The first class is the smallest, and the 
second is the largest. Taking the three classes in order of 
their size, their numbers are approximately proportional to i, 
4, 6, as the lists given below suggest. It was common usage 
to divide arithmetic into theoretic and practical. For example, 
Trenchant, in the beginning of his arithmetic, defined these 
divisions thus : 3 " Theory is the speculation through which 
one becomes acquainted with the property of numbers. Prac- 
tice is the performing of the operations which arise from such 
knowledge and speculation." 

As would be expected, practical arithmetic was about the 
same in its general features wherever found. The exceptions 
are due to local influence or to eccentricity of authorship. For 
example, Champenois drew his problems mainly from military 
affairs. Thus : 

r ' The Commissary-General had four stewards ; the first 
had 7,836 loaves of bread, the second 6,342 loaves, the third 
5,424 loaves, and the fourth 6,398 loaves. The question is: 
How much bread did they have altogether?' " 4 

"A squadron contains on the front 312 men and on the 
side 232. The question is : ' How many men are there in a 
squadron?' " 5 

1 Tonstall (1522), Paciuolo (1494), Cardan (1539), Noviomagus (1539), 
Tartaglia (1556), Gemma Frisius (1540), Riese (1522), Ramus (1567)*- 
Trenchant (1571), Champenois (1578), Unicorn (1598). 

2 Borgi (1484), Calandri (1491), Widman (1489), Cirvelo (1505), Ru- 
dolff (1526), Kobel (1531), Baker (1562), Raets (1580), Van der Schuere 
(1600). 

3 Trenchant, L'Arithmetique (1578). " La Theoreque eft la f peculation 
par laquelle Ton vient a connoetre la propriete de leur fuget. Et la pra- 
tique, eft Foperation & effet qui prouient de telle connoefance & fpectK 
lation." Fol. A 4 verso. 

4 Champenois, Les Institutions De L'Arithmetique (1578). "La Commis 
general des viures a quatre prouif eurs de pain. Le premier a 7838 pains : 
le fecond, 6342 : le troif iefme, 5424 : & le quatriefme, 6398. Lon demande 
combien il y de pain en tout." Page 10, fol. Bv verso. 

5 Champenois, Les Institutions De L'Arithmetique (1578 ed.). "Vn efca- 



THE ESSENTIAL FEATURES i2 q 

"A captain with 4,000 soldiers is besieged in a fortress by 
the enemy for seven months; they have food for five months 
and are without hope of obtaining- any during the period of 
the siege, which is seven months. The question is : ' How 
much should the captain diminish the rations of the soldiers 
that the food may last through the time of the siege, which 
is 7 months ?' " 

Another writer who used peculiar problems was Suevus. 
His interest inclined to history. In the dedication of his 
arithmetic he named the following applications : x "To reckon 
feast days of the church, also many mysteries and secrets of 
the church. For use in schools, which Cicero, that learned 
pagan, called the foundation of the whole republic, as all other 
arts are learned so much better through arithmetic. Also in 
the army; among merchants; among manual workers, as 
artists, goldsmiths, mint masters, watchmakers, painters, 
builders, masons and others; and in housekeeping." 

The subject-matter of the book presents a most remark- 
able collection of historical material. Some of the problems 
illustrate a tendency not uncommon at that time to correlate 
religious teaching with school instruction. Under Numera- 
tion the first example is : " The number of years from the be- 
ginning- of the world to the birth of Christ, our Saviour, and 
the Incarnation was : 

3970. 

" That is, Three thousand nine hundred seventy years. 

" That was the time decided upon when God promised to 
send his Son. This promise he had fulfilled (Galat. 3), by 
which we may know his truth and uprightness, and putting- 
aside all sorrow and doubt, we may sing cheerfully with the 
dear David from the 33d Psalm, and we may say : ' The 

dron contient en front 312 hommes, & en flanc 232. Lon demand e eombien 
il y a d'hommes en l'efcadron." Page 27, fol. Cvi recto. 

" Vn Captaine auec 4000. foldats eft aff iege en vne forteref f e, de 
l'ennemy pour 7. mois, & n'ont de viures que pour 5. mois, & sans efper- 
ance d'en pouuoir recouuir durant le temps de raffiegement qui eft 7. 
mois? Lon demande combien le Captaine doit apetiffer la penfion du 
foldat, afin que le viure puiffe durer le temps de raffiegement, qui eft 7- 
mois." Page 83, fol. Gij recto. 

1 Suevus, Arithmetiea Historica (1593 ed.). 



130 SIXTEENTH CENTURY ARITHMETIC 

word of the Lord is true, and that he hath promised will he 
surely fulfil." x 

Other examples under Numeration are: 

"According to the statements of Theodore Bibliander, the 
cost of building Solomon's temple was 13,695,380,050 
crowns." 2 

" The yearly cost of maintaining the wars of the Emperor 
Augustus, especially in holding the Roman borders, was 12,- 
000,000 crowns." 

" The annual income of King Ptolemy Auletes was 7,500,- 
000 crowns." 3 

Addition finds application in determining the age of Methu- 
selah : 

"Methuselah was 187 years old when he begot Lamecli; 
after that he lived 782 years. What was his age? Ans. 969 
years." 4 

Division is applied thus : 

" In his thirty-fourth Book, Livy informs us that 1,200 
Gallic prisoners were released with 100 talents. The question 
is: ' How much should each receive?' " 5 

1 Suevus, Arithmetica Hdsrtorica (1593). "Die Jarzal von anfang der 
Welt/ biss auff Ohristi vnsers Heylandes Geburth. vnd Menschwerdung. 

3970. 

" Das sind : Drey tains-end/ neun hundert vnd' >siiebentzig Jar. 

" Das ist die bestimpte zeit/ darin Gott seinen Son zu senden verheissen/ 
auch seine zusage kreffigerfullet hat/ Galat. 3. daraus wir seine Trew vnd 
Wanheit kennen lernen/ vnd wir alien kumimer vnd zwaiffel/ mit dem 
lieben David aus dem 33. Psalm getrosit isingen und sagen mugen : Des 
Herrn Wort ist warhafftig/ vnd was er zusagt/ das foelt er gewiss." Fol. 
Aij verso. 

2 " Des Tempels Salomonis vnkosten zu bawen/ nach des Theodori Bib- 
liandri verzeichnis. 13695380050 Cronen." Fol. Aij verso. 

" Des Keysers Augusiti Jarlich Kriegs vnkosten/ sondierlich des Rom- 
ischen Reichs Gretzen zu halten. 12006000 Cronen." Fol. Aiij recto. 

3 " Des Konigs Ptolomei Auletis Jahrliohs Einkommen. 7500000 Cro- 
nen." Fol. Aiij recto. 

4 Methuisalem war hundert vnd sieben vnd achtzig Jahr alt/ vnd zeugete 
Lamech/ vnd lebete darnach sieben hundert vnd zwey vnd achtzig Jahr. 
Wie gros ist denn sein gantzes Alter geworden ? Antwort, neun hundert 
und neun vnd sechzig Jahr." Page 22. 

5 "Liuius Lib. 34 meldet : 'Das zwolff hundert Welche gefangene Kriegs- 
leute mit ihundert Talentis sind ausgeloset worden.' " 

" 1st die Frage : ' Wie viel fur eine Person gegeben sey ?' " 



THE ESSENTIAL FEATURES 1 ^ 1 

The following problems occur under the Rule of Three : 

" In the 7th Chapter of his 12th Book, Pliny tells us that a 
pound of black pepper was bought for 4 denarii — that is, for 
a half Taler. Here is the question: ' If 3! pounds cost 12! 
denarii, what will 14! pounds cost?' " 1 

" Martial (the poet) informs us that an amphora of wine 
was sold for 20 asses, that is, for 2 denarii, which is as much 
as a quarter o>f a Taler. The question is : 'At this rate how 
much should a Roman sextarius cost, if it takes 64 sextarii to 
fill a Greek amphora?' " 2 

In the early printed arithmetics, as in those of Borgi 3 and 
Calandri,* the number of problems is meagre, each problem 
usually being followed by its solution. In the arithmetics of 
a date later than 1525 problems for practice are numerous. 
The explanation is probably to be found in the fact that the 
cost of paper and printer's composition rapidly decreased. 

Although, as we have said, applied arithmetic was gener- 
ally thought of as a department by itself, certain applications 
were distributed among the simple operations. The most im- 
portant of these are the problems of denominate numbers (pp. 
77-85 of this article), which were real applications and not 
mere exercises in manipulating symbols often found in modern 
arithmetics. But, passing to that department commonly called 
practical arithmetic by sixteenth century authors, we find that 
it was generally composed of a list of rules of operations under 

1 Suevus, Arithmetica Historica (1593). " PLinius Lib. 12. Cap. 7. mel- 
det: Das man ein Pfund sdiwartzen Pfeffer vmb vier Denarios gekaufft 
habe/ das ist vmb einen halben Taler. 

" Hier ist die Frage : Wenn 'drey Pfundt/ vnd drey viertel eines Pfundes 
vmb zwolff Denar/ vnd vier Funffiel eines Denarij gekaufft wurden : Wie 
tewr vierzehen Pfund vnd zwey Driittel eines Pfundes im Kauff sein wur- 
den?" Page 250. 

2 Suevus, " Martialis meldet, Das ein Amphora Wein sey vmb 20. Asses 
verkaufft worden/ das ist vmb 2 Denar/ so viel als ein Ort eines Talers. 
1st die Frage : Wde thewr ein Romisch Sextarius oder Nossel/ deren vier 
vnd sechitzig auff ein Griechische Ampihoram geihen/ zu rechnen sey?" 
Page 256. 

3 Borgi, Arithmetica (1488 ed.). 
4 Calandri, Arithmetica (1491), 



132 SIXTEENTH CENTURY ARITHMETIC 

which are grouped the corresponding problems of business 
concern. 

The following may be taken as a typical category : 

Rule of Three (Two and. Five). Exchange and Banking. 

Welsch Practice. Chain Rule. 

Inverse Rule of Three. Barter. 

Partnership (with and without time). Alligation. 

Factor Reckoning. Regula Fusti. 

Profit and Loss. Virgin's Rule. 

Interest, Simple and Compound. Rule of False Assumption, or False 

Equation of Payments. Position. * 

Besides these more general rules, the following were often 
added : 

Voyage. Rents. 

Mintage. Assize of Bread. 

Salaries of Servants. Overland Reckoning. 

Gemma Frisius recognized the dependence of most of the 
rules given in the above list upon the Rule of Three: * " From 
this one rule, which in fact may be called ' The Golden Rule,' 2 
grow many different rules or methods of work, as the 
branches of a tree grow from its trunk, so much so that it 
has place in nearly all questions, and all canons lean upon it 
as a foundation, or base, one of which is the Double Rule, 
which you will understand from the following example." 

The Rule of Three 3 was the method of simple proportion. 

1 Gemma Frisius, Arithmeticae Practicae Methodus Faeilis (1575 ed.). 

De Regulis vulgaribus. 
" Ex una hac regula (quam vere auream licet appellare) multae diuer- 
saepj regulae, siue Canones operandi tanquam rami ex trunco oriuntur, 
adeo vt in omnibus fere quaestionibus locum habeat ac omnes Canones 
hinc innitantur, tanquam fundamento seu basi, quarum vna est regula 
duplex, quam ex tali exemplo intelliges." Fol. D g recto. 

2 Jacob, Rechenbuch auf den Linien und mit Ziffern/ (1599 ed.). "So 
viel hat ich von den progreffionen erzehlen und fetzen wollen/ Folgt/ 
ferrner die Regel De tri/ von etlichen/ Proportionum, auch fonften Aurea 
Mercatorum genannt." Fol. D Q recto. 

3 Aryabhatta (c. 500 A. D.) used Rule of Three. See Rodet's Lecons 
de Calcul d'Aryabhata, 



THE ESSENTIAL FEATURES 133 

Its explanation was sometimes expressed in general terms by 
the writers of arithmetic, as in Heer's Arithmetic : * 

" I am composed of three parts ; always place the question 
last; whatever number is like the question put in the first 
place. Multiply together the last and middle numbers and 
divide the result by the first number. The quotient will be of 
the same kind, or denomination, as the middle number. Thus 
is the question solved." 2 Such generality, however, was un- 
usual, the solution of a typical example ordinarily served as a 
guide without the aid of a general theory or formula. 

This statement of the rule was given by Borgi : 3 
"A Rule Pertaining to Trading." 

" Three quantities are known to find the other. Multiply 
the second by the third and divide the result by the first." 

" Example. The three numbers are 2, 3, 4. 

3X4= 12 12 -f- 2 = 6." 

1 Heer, Compendium Arithmeticae (1617 ed.). 

" Von dreyen bin ich zufamm gef etz 
Die Frag fetz alle mal zu letzt 
Vnd was die Frag fur Namen hat 
Das ordne an die vorder ftatt 
Das hinder vnd mitler Multiplicir. 
Was komt durchs vorder Dividir 
Der Quotient bringt dir zur frift 
Den Nam/ fo mitten geftanden lift 
Damit ift der Frag auffgeloft du wift?" 

Fol. Avii verso and Aviii recto. 

2 It is clear from this rule that, if the unknown term had been written, 
it would have taken the fourth place in the proportion, whereas in 1 the 
modern form it takes the first place. 

3 Borgi, Arithmetica (1540 ed.). 

" Como fi procede in tutte rafon merchadantefche per ditta regola." 

" Altro non ci refta f e non a ueder in che modo per la precedete regola 
fe die proceder in el far delle rafon merchadantefche, commciando in 
quefto modo, fel te fuffe detto, fe .2. val .3. che valera .4. prima metti 
quefte tre cofe vna drieto a laltra, ooe 2. val .3. che valera .4. prima metti 
quefte tre cofe vna drieto a laltra, cioe .2. 3. & .4. fi eome tu vedi, poi 
moltiplica la feconda an la terza, cioe .3. via .4. e fara .12. el qual .12. 
parti per la prima, cioe per .2. & in fira .6. e tanto^ val el .4. adonque achi 
ti diceffe fe .2. val. 3. che valera 4. tu hai a rifponder che 1 val. 6. 

" Se braza .3. de tela val fol .15. che valera el brazo. 

" Se braza .4! de tela val fol. 17. che valera braza. 8." Fol. E g recto. 



I 3 4 SIXTEENTH CENTURY ARITHMETIC 

Problems: i. If 3 braza cost 15/7. what does 1 braza cost? 

&|3 P*S b \ I 
2. If 4^ braza are worth p 15, what will 8 braza be worth? 

b\ 4 y 2 pi 7 b\s 

Problem 1, of course, is a direct case of division. Problem 
2 is the usual type of proportion. It was not unusual to pre- 
face problems like the second by some like the first. 

It has been explained on pp. 1 19-120 of this chapter that 
proportion, came to have its present meaning in the sixteenth 
century. Consequently one would expect to find some use of 
proportion in solving problems and its relation to the Rule of 
Three. This connection is supplied by Buteoi in the following 
treatment : 

" Three numbers being given, to find a fourth proportional 
number, called the Rule of Three." 

" By many, indeed, it is called the Regula trium, or, as a 
certain contemporary foreigner wrote it, regula de tri. By 
others it is called ' The Rule of Four Proportional Num- 
bers/ " x 

Ramus also combines the rule with proportion, for after 
applying it to problems involving integers and fractions in the 
usual way, he gives the following chapters : 

Chap. VII. Golden Rule requiring Antecedent Proportion. 

A typical problem is : " The lion on a fountain had 4 pipes, 
of which the first fills the pool below in 24 hours, the second 
in 36 hours, the third in 48 hours, and the fourth in 6 hours. 
If they flow simultaneously, in how many hours will they fill 
it? Add in turn the four ratios, 1 pool to the 4 periods of 
time, and the total will be 37 pools and 144 hours, antece- 
dents of the proposition. Since 37 pools are filled in 144 
hours, therefore 1 pool is filled in 3ff hours." 2 

1 Buteo, Logistica, Quae & Arithmebica vulgo dicitur (1559). "A multis 
fiquidem dicitur regula trium, vel ficut quidam Barbarus fcripfit tempore 
noftro, regula <ie tri. Ab aliiis regula quatuor proportionalium." Fol. g 4 
recto, p. 104. 

2 Ramus, Arithmeticae Libri duo. Liber II, Cap. VII. 

"Leo f otitis 4 fistulas 'habet, quarum prima implet lubjectum lacum 24 
horis, fecunda 36, tertia 48 quarta 6: fi fimul fluant, quot horis implebunt? 



THE ESSENTIAL FEATURES ! 35 

Chapter VIII. Concerning Reciprocation. 

A typical problem is : " When a measure of wheat is sold 
for 5 aurei, a loaf of bread weighs 4 unciae; when, however, 
it is sold for 3 aurei, a loaf of bread will weigh 6f unciae." 5 

Chapter IX. Composite Proportion through Addition. 

Chapter X. Alligation. 

Chapter XI. Composite Proportion through Multiplica- 
tion only. 

"A piece of tapestry 2J ells long and 2 ells wide was sold 
for 50 libelli, therefore a piece of tapestry of the same quality 
1 ell long and f of an ell wide will sell for 1. 8 s. 6 d. 8." 2 

Chapter XII. 'Composite Proportion through Multiplica- 
tion and Addition. 

Chapter XIII. Continuous Proportion for Finding the 
Smallest Term in a Given Series. 

In the fifteenth century the Italians originated a modifica- 
tion of the Rule of Three for certain problems involving de- 
nominate numbers. This method was first published in Ger- 
many by Schreiber (Grammateus) in 1518 3 under the name 
of Welsch Practice. At that time the people of southern 
France and northern Italy were often called Welsch by the 
Germans, whence the name given to the process. Italian 
writers refined the processes in arithmetic in many ways, 

Adde rurfum quatuor rationes 1. lacus ad quadruplex tempus, tota ratio 
eri't z7 lacuum 144 boras anteoedens proposition's. Die igitur Z7 lacus ini- 
plenitur 144 ihoris, ergo 1. lacus impletur horis 3^5. fie: 

1. 24 

1. 36 

1. 48 

1. 6 



37. 144- 1. 3ff" 



1 Ramus, Arithmeticae Libri duo (1577 ed.). " Cum modius tritici vaenit 
5 aureis, turn panis eat 4 unciarum: ergo cum vaenit 3, panis erit unciarum 
6 2 / 3 ." Page 59. 

2 Ramus, "Aulaeum longum ulnas 2 & -1, latu 2 emitur 50 libellis : ergo 
tapetum ejusdem generis alterum longum ulnam 1, latum -| emetur 1. 8. 
s. 6. d. 8." Page 73. 

3 Schreiber, Ayn new Kiinstlich Buch welcher gar gewifs vnd behend/ 
lernet nach gemainen regel D'etre/ wefochen practic/ (1518 ed.), fol. D 1 



! 3 6 SIXTEENTH CENTURY ARITHMETIC 

which led to the custom of designating- any ingenious opera- 
tion originating with them as Welsch Practice. Thus, the 
name became widely used in the broader sense, occurring in 
the titles of many arithmetics of the sixteenth and seventeenth 
centuries. 

The technical difference between the Rule of Three and 
Welsch Practice in the original narrow sense may be seen 
from two examples from Riese. 

i. Problem. " If i pound costs 3 groschen 9 denarii, how 
much will 3 centner 2 stein 7 pounds cost ?" 1 

Solution by the Rule of Three : 

1 : 3 groschen 9 denarii = 3 centner 2 steins 7 pounds : 
( ). Since 1 centner =110 lb., 1 stein = 22 lb., then 3 
cent. 2 stein 7 lb. = 381 lb. Reduce the 3 gr. 9 d. to denarii. 
The result is 45 d. 

Then 1 145 = 381 lb. : ( ) 

Reduce 381 X 45 d. to florins, groschen, and denarii. 

2. Problem. If 5! lb. cost 32 fl. 13 gr. 12 d., what will 
47 lb. 25 lot cost? 

Then 5! : 32 fl. 13 gr. 12 d. = 47 lb. 15 lot: ( ). 

He now has to multiply the means together as in the Rule 
of Three, but instead of reducing each compound number to 
one denomination he performs the work as follows : 

S| lb. kosten 32 fl. 13 n gr. 12 cf. was 47 lb. 25 lot? 
6 16 l\h. 



35 194 A- 22 n gr. 8 



1358 fl. 10 

776 



fl. 1 



2 
15 fl. 20 n gr." 

— 22 gr., mal. 47 



15 tl. 20 n gr.-| 
15 " 20 " 
3" 4 " J 

] 



97" 11 

48 " 20 " 9 ^. I - Preis fur 25 Lot 
6 ' ' 2 " io| 
9304 fl. 18 n gr. i\ <5. Summa 



5) 

i860 " 27 " 11^ " 

7) 

265 fl. 25 n gr. 6%U 6 - 

Riese, Rechnung auff der Linien vnd Federn/ (1571 ed.), Unger, p. 92. 



THE ESSENTIAL FEATURES i^y 

Explanation, i. Multiply the terms of the first ratio by 6 
to avoid fractions. 

Then, 35 : 194 fl. 22 gr. = 47 lb. 15 lot : ( ). 

The equivalents here are 1 fl. = 30 gr., 1 gr. = 18 d., and 

I lb. = 32 lot. 

2. Multiply 194 fl. by 47. The result is 1358 

776 

3. Multiply 22 gr. by 47. To do this separate 22 gr. into 

10 gr. + IO §T r - + 2 g T -> which equals Yz fl. + Yz fl. + Ys 
of Yz A., and multiply each by 47. The result is 15 fl. 20 gr. 
-f- 15 fl. 20 gr. + 3 fl. 4 gr. (Note that 10, 10, 2 are parts 
of 30 gr. (= 1 fl.) expressed by unit fractions.) 

4. Multiply 194 fl. 22 gr. by 25 lot. To do this separate 
25 lot into 16 1. + 8 1. + 1 1. = Y* lb. + Yt. of Y* lb. + Ys 
of Y^ of Y2 lb. and multiply 194 fl. 22 gr. by each term. The 
result is 97 fl. n gr., 48 fl. 20 gr. 9 d. by taking Y of the last 
result, and 6 fl. 2 gr. ioj^ by taking Y of the last result. 
(Note again the unit fractions.) 

5. The partial products are now added and divided by 5 
and 7, or 35. 

It is the ingenious method of separating the multipliers in 
order to avoid reduction to one denominator, as in the solu- 
tion by the Rule of Three, that constitutes the characteristic 
feature of Welsch Practice ; a good example of how the short- 
est process may become the longest. 

Problems then solved by the Rule of Three are now solved 
by unitary analysis or by the equation, and Welsch Practice 
has lost its virtue on account of the comparative simplicity of 
the modern denominate number systems. It is interesting to 
note that Welsch Practice in its early form exactly coincides 
with our present unitary analysis. This may be seen by an 
example from Calandri : x " If a cogno of wine is worth 33 y 

II /? 4 6, what are 5 barili worth?" 

1 Calandri, Arithmetica (1491 ed.). "El cogno del uino uale 33 y 

11 P 4 s che uarranno 5 barili." Fol. 28 verso. 



SOLUTION 






33 — 


ii - 


- 4 


5 


3 — 


7 - 




"'* 


15 








i - 




15 - 


— o 


o - 
o - 




o - 

o — 


— 5 






3 



138 SIXTEENTH CENTURY ARITHMETIC 

EXPLANATION. 

10) 33 — 11 — 4 — - 5 The top line is the statement 

of the problem, namely, 10 barili 

: 33 if 11 P 4 * = 5 barili : 

( ). This is evident, since 1 

cogno =10 barili. The second 

line is found by dividing the first 

16 y 15 p 8 rf two terms by 10. This gives the 

cost of 1 barili. The other lines 

are the results of multiplying the denominators of the second 

line by 5, the relations being 20j8= 1 Jf, 12 6 =1/?. 

Thus, the whole solution is briefly, if 10 barili cost 13 ft 
11 p 4 s, 1 barili costs T V of this. 5 barili cost 5 times the 
latter result. This process we call unitary analysis. 

In the Double Rule of Three, also called the Rule of Five, 
are recognized the problems o*f compound proportion. Five 
quantities are given to find a sixth. A typical example from 
Gemma Frisius is: 

" If 4 aurei must be paid for transporting 20 pounds of 
merchandise 30 miles, how much must be paid for transport- 
ing 50 pounds 40 miles ?" * 

The Rule of Two was the process of dividing the product 
of two numbers by the sum, and was applied to problems 
whose solution could thus be obtained. In the Treviso book 2 
it is applied to certain courier problems. 

Inverse Ride of Three 
The Inverse Rule of Three is explained by Baker thus : 

" Of the Backer Rule of Three." 
" The backer rule of three is so called because it requireth a 
contrarye working to that, which the Rule of three direct 
doeth teach e, whereof I have no we treated." 3 

1 Gemma Frisius, Arithmeticae Practicae Methodus Facilis (1581 ed.)- 
" Pro 20 Libris cuiiufui-s mercis, aductis per 30 milliaria, foluendi lunt 4 
aurei, quantum pro 50 Mb. aducctis per 40 milliaria ?" Fol. E g verso. 

2 See p. 9, bibliographical note. 

8 Baker, The Well Spring of Sciences (1580 ed.), fol. Gva recto. 



THE ESSENTIAL FEATURES I39 

Example. "If 15 shillinges worth of Wyne will serue for 
the Ordinary of 46 men when the Tonne of wyne is woorth 
12 pounds: for howe many men will the same 15 shillinges 
worth of wine suffice; when the ton of wine is woorth but 8 
pounds." 

Partnership 

The first kind, simple partnership, was concerned with 
dividing a gain or loss proportional to a given set of numbers. 
The following is an example from Suevus prefaced by an in- 
troduction resembling a Herbartian Preparation : * 

Introduction. " In the first chapter of the Prophet Jonah 
we read how God enjoined upon the Prophet to proclaim to 
the Ninevites his righteous anger against their sins; this 
Jonah refused to do, and for this reason betook himself to a 
ship and departed with the sailors. But Almighty God ar- 
rested the progress of the Prophet Jonah by a mighty tempest 
which greatly frightened the sailors. As soon as they had 

1 Suevus, Arithmetica Historica (1593). " Im Propheten Jona Cap. 1. 
lesen wir/ wie gott der herr dem Propheten aufferleget hat/ den Niniuiten 
seinen gerechten Zorn wider ihre Sunde zuuerkundigen/ des sich <ier 
Prophete gewegert/ vnd sich derhalben auff ein Schiff begeben hat/ vnd 
mit <ien Schiffleuten dauon gefahren ist/ aber der Allmachtige gott hat 
den Propheten Jonam, duroh einen grossen Sturmwind auff dem Meer 
arestiret vnnd auffgehalten/ daruber die Schiffleute sehr erschrocken sind/ 
als bald das Schiff zu/ leichtern/ etlich Gerethe aussgeworffen/ audi 
darumb (sonder zweiffel aus sonder schickung gottes) das Loss geworffen 
haben/ zu erkundigen/ umb wen es doch musse zu thun sein/ und weil 
das Loss den Propheten Ionam getroffen/ hat er sich willig darein begeben/ 
das sie ihn aus dem Schiffe ins Meer gesturtzet haben/ welchen als bald 
ein grosser Wallfisch auffgefangen vnd verschlungen/aber noch dreyen 
tagen vnd nachten wieder zu Rande vnd Lande gebracht hat/ das er nach 
dem befehl das Herrti den Niniuiten die Busse gepredigt hat. 

" Dauon wollen wir auch ein nutzlich Exempel nemen. 

" Wan vier Kauff leuthe ein Schieff mit Guttern beladen hatten/ 

" 1. Einer mit vier und funfftzig Lasten : 

"2. Der ander mit zwey vnd Siebentzig Lasten: 

" 3. Der dritte mit Hundert vnd vier vnd zwantzig Lasten : 

" 4. Vnd der vierde mit Hundert vnnv Funfftzig Lasten : Jeder Last 
auff 12 Tonnen zu rechnen/ dauon die Sohieffleute inn grossem vngewitter 
das Schiff zu Leichtern/ haben Sechs Last vnd vier Tonnen auswerffen 
mussen. Ist die Frage: Wie viel ein jeder Kauff man in sonderheit habe 
schaden Leiden mussen." Pages 339-340. 



140 



SIXTEENTH CENTURY ARITHMETIC 



thrown over some of their wares to lighten the ship, they cast 
lots (without doubt by the special Providential arrangement 
of God) to find for whose sake the calamity had to happen, 
and because the lot fell to the Prophet Jonah, he voluntarily 
gave himself up that they might cast him from the ship into 
the sea : as soon as they did this a great fish seized and swal- 
lowed him. After three days and nights, however, he was 
brought to the shore again that he might preach of repentance 
to the Ninevites." 

Problem. " From this we also wish to take a useful ex- 
ample : 

" Four merchants loaded a ship with goods ; the first fur- 
nished 54 Lasten, the second J2 Lasten, the third 124 Lasten, 
and the fourth 150 Lasten. Each Last is to be considered as 
12 tons. In a violent storm the sailors had to throw over- 
board 6 Lasten and 4 tons to lighten the ship. The question 
is : ' How much must each mercant share in the loss ?' " 

The second kind, Partnership with Time, may be illustrated 
by the following from Riese : 1 

" Three men formed a partnership. The first gave 20 

florins for 4 months, the second 24 florins for 3 months, and 

the third 40 florins for 1 month. They made 101 florins; 

how much belonged to each ?" 

Ans. 1st. 42 fl. 1 fi 8 heller. 
2d. 37 fl. 17 fi 6 heller. 
3d. 21 fl. 10 heller. 

80 

192 101 fl. 72 

40 

1 Riese, Rechnung auff der Linden und Federn/ (1571 ed.). "Item/ 
Drey machen ein Gefelfcihafft alio/ Der erft legt 20 floren/ 4 monat/ Der 
ander 24 floren drey monat/ Vnd der dritte 40 floren ein monat/ Haben 
101 fl gewunnen/ wie viel geburt jgLichem? Facit dem erften 42 fl/ 1 &/ 
8 heller/ Dem andern 37 fl/ 17 &/ 6 heller/ Vnd dem dritten 21 fl/ 10 hi'. 
" Machs alio/ multiplioir jglichs Geld mit feiner zeit/ Summir/ wird 
d'em Theiler/ Vnd fetz darnach in maffen/ wie du oben gethan haft/ 
Stehet alfo." Fol. Hviiii recto. 

80 

192 101 fl 72 

40 



THE ESSENTIAL FEATURES 141 

The problems in partnership reflect real conditions of that 
time, although they seem artificial now, for there was a pop- 
ular prejudice among Christians against taking interest, and 
usury laws were made to prevent it. Hence, merchants 
pooled their money in enterprises for various periods, often 
very brief, and shared the profits in proportion to the amounts 
invested and the times for which they were furnished. 1 

The problems concerning pasturage (pasturing partner- 
ship 2 ) are relics of the days of commons and shepherds. 
Contractors rented large sections of the estates, which had 
been farmed out to them by the state, to the owners of stock, 
and these paid in proportion to the numbers in their herds. 3 
Such problems may be found in a few arithmetics at the 
present time. 

Tonstall gives a third case o-f partnership in which, while 
different times intervene, money is drawn out also. 

Example. " Four merchants formed a partnership for two 
years. The first contributed 30 aurei at the beginning, and 
after 8 months drew out 10 of them. At the beginning of the 
twentieth month he contributed 12 aurei to the partnership. 
At the beginning the second contributed 24 aurei, and at the 
beginning o>f the sixteenth month he contributed 14 more 
aurei. The third contributed at the beginning 20 aurei, at the 
beginning of the seventh month he withdrew all his money, 
and at the beginning of the eighteenth month contributed 16 
aurei. The fourth at the beginning of the seventh month con- 
tributed 18 aurei, and at the beginning of the fourth month 
after drew out 9 of these; again in the seventeenth month he 
added 15 aurei to the business. How should a gain of 100 
aurei be divided among them?" 4 

He gives the following solution with a careful explanation : 

1 W. Cunningham, The Growth of English Industry and Commerce 
during the Middle Ages (London, 1896), p. 364. 

2 Ortega's Arithmetic (1515 ed.), fol. 762, has the expression " com- 
pagnia pecovaria." 

* Ramsey, Manual of Roman Antiquities (London, 1901), p. 548. 
4 Tonstall, De Arte Supputandi (1522), fol. Z 2 verso. 



620 




3sm 


558 


100 


31m 


252 




I4HI 


3i8 




i8AV 



I 4 2 SIXTEENTH CENTURY ARITHMETIC 



17480 



Factor Reckoning corresponds to the modern topic of com- 
mission, as shown by the following problems from Baker : 

"A marchant hath delivered to his Factor 1200 li. to gouerne 
them in the trade of marchandife, upon fuch condition, that 
he for his feruice fhall have the i of the gaine, yf anything 
be gained, and he fhall beare the i of the loss if any thinge be 
loste : I demaunde for how much his person was esteemed." 1 

"A marchante hath delivered unto his Factor 1200 l'i and 
the Factor layeth 500 Ti and his person. Nowe, because hee 
layeth in 500 li. and his person, it is agreed between them y 
he shall take f of the gaine: I demaunde, for how much his 
person was esteemed ?" 2 

Profit and Loss 

Problems of this kind were often unclassified and used as 
applications of the Rule of Three. But some writers grouped 
them under a separate title, thus setting the precedent fol- 
lowed until the present time. 3 

A type example and the plan of solution is seen in the fol- 
lowing from Riese: 

"A man bought a centner of wax for i6| florins. How 
many pounds will he sell for 1 H. if he wishes to make 7 H. on 
100 florins? Answer, 5 pfund 29 loth 3 quintle 2 pf. gewicht 
o*H* heller." 4 

(Reckon first how much wax was bought for 100 florins, 

1 Baker, The Well Spring of Sciences (1580 ed.), fol. Wii recto. 

2 Baker, The Well Spring of Sciences (1580 ed.), fol. Wv recto. 

3 Sfortunati's Arithmetic (1545 ed.), fol. 47 recto, has "di guadagni e 
perdite." 

4 Riese, Rechnung auff der Linien und Federn/ (1571 ed.). "Item/ Ein 
Centner Wache fur 16 floren/ 3 ort/ Wie viel pfund komen fur 1 £1/ fo 
man an 100 gewinnen wil 7 floren? Facit 5 pfund/ 29 loth/ 3 quintle/ 
2 gewicht/ heller/ vnd |~f-|-^- teil. Machs alio rechne zum erften 
wie viel Wachs fur 100 floren kompt/ Ah denn addir die 7 floren zu 100/ 
vnd fprich/ 107 floren geben fo viel Wachs/ als hierin 634|-| lb/ Was 
gibt 1 fl? Brichs/ Itehet alio." 

6741 40000 tb 1 fl. Fol. Evii verso. 



THE ESSENTIAL FEATURES I43 

then add 7 florins to 100, for which he will sell the same 
amount of wax, or 634ft lb.) How much does he sell for 
1 A? 

6741 — 40,000 lb. — 1 a. 

It was common to speak of the gain per hundred, or per 
cent, as it is now, for per cent always meant a rate in that 
period. Thus, in Rudolff: 

"A piece of velvet cost 36 flo.; it contained 15! ells, for how 
much should 6 ells be sold so as to gain 10 florins on a 
hundred? 

Gain was also reckoned with time. The first example in 
Tonstall is : 

"A merchant gained 5 aurei in 3 months from 70 aurei. 
At this rate what would he gain from 70 aurei in 13 
months ?" x 

The other cases treated by Tonstall were : To find the time 
when the gain is given. To find the gain from a larger 
amount when the gain on a small amount is given. To> find 
the time in which a gain greater than the money invested can 
be found. 

Simple Interest 

Van der Schuere began the subject thus: 2 Simple interest 
is much like gain and loss with time, so that if you can work 
one subject, you can easily understand the other. 

"A man placed 100 L at simple interest for 4 years at 6^4% ; 
how much did he have at the end of the time?" 3 

The rates varied from 6 per cent to 12 per cent, although 
there were exceptional extremes. 4 

1 Tonstall, De Arte Supputandi (1522). "Mercator ex avreis septva- 
ginta per menfes tres lucri fecit quinq?. quatum" hicri tredecim menfibus ex 
aureis feptuaginta obueniet?" Pol. a^^ verso. 

2 Van ider Schuere, Arithmetica, Oft Reken = const/ (1624 ed.). 

"Den simp'len Int'rest, is, Winst end Verlies met Tijdt, 
■Ghelijckend' een groot deel, dus van het werck subijt 
Sul-dy veel haest verstant ghecrijghen door u vlijt." 

Fol. Pv recto. 

3 Van der Schuere, Arithmetica (1624 ed.). " Een en gheeft op Interest 
100 L voor 4. Jaer/ om daer voor te hebben simpeleln Interest/ teghen 
6% ten 100 t's Jaers/ Hoe veel ontfang hy dan ten eynde des tijdts." Fol. 
Pv recto. 

4 Raets mentions 14%, and Trenchant 10% (1578 ed.), p. 300. 



144 



SIXTEENTH CENTURY ARITHMETIC 



Jean solved problems of interest from a table. This ex- 
ample -will illustrate: 

" I wish to find the interest on 720 livres at 16 deniers per 
livre. On line 16 I search for the sum, and when I find it I 
refer to the number at the top of the column, where I find 45 
which is the interest on 720 livres. " 1 

Trenchant gives the following interest table with interest at 
12 per cent on sums from 10,000 livres to 1 sou. 2 

Principal. 

9 1'. 
8 

7 
6 

5 
4 
3 
2 

1 r 
19 f 
18 

17 
16 
15 
14 
13 
12 
11 
10 
9 

8 
7 
6 
5 
4 
3 
2 
1 

The following is an example worked from the above table : 
" To find the interest on 16,097 livres 8 sous." 

1 Jean, Arithmetique (1637 ed.). " Ie veux itirer Tinterest au denier 16 
de la .S'omme de 720 liures : le cherche done ladite somme dans la ligne 16, 
& l'ayant trouuee, ie regarde directement au des-sus en la ligne capitale, ou 
ie trouue 45, qui est 45 Mures de rente que donnent lesdites 720 liures." 
Fol. Aiiij recto. 

2 Trenchant, V Arithmetique (1578 ed.), fol. M 6 recto. 



Principal. 


Interest. 


10000 r. 


1200 r. 


9000 


1080 


8000 


960 


7000 


840 


6000 


720 


5000 


600 


4000 


480 


3000 


360 


2000 


240 


1000 


120 


900 


108 


800 


96 


700 


84 


600 


72 


500 


60 


400 


48 


300 


36 


200 


24 


100 


12 1\ r. 


90 


10 — 16 


80 


9 — 12 


70 


8—8 


60 


7—4 


So 


6 — 


40 


4 — 16 


30 


3 — 12 


20 


2—8 


10 


1 — 4 



Interest. 




1 1\ 1 f. 


7jd- 


19 — 


2i 

O 


16 — 


9f 


14 — 


4* 


12 — 




9 — 


n 


7 — 


2 t 


4 — 


9| 


2 — 


4* 


2 — 


3A 


2 — 


.'If 


2 — 


°tt 


1 — 


"A 


1 — 


9« 


1 — 


8 A 


1 — 


m 


1 — 


sA 


1 — 


3f* 


1 — 


2 f 


•r 


24 




U ^F 





"if 





I0 A- 




816 




°2T 





7* 




Cl9 







4A 





2 ff 





4* 



THE ESSENTIAL FEATURES 145 

(Soit maintenant qu'il faille fcauoir les interests de 16097 
liures, 8 fouz de principal.) 

10000 1' 1200 1' of den. 

6000 720 

90 10 16 

7 

16 9f 

0—8 r 1114 

2o 



1931 r 13 r 



'25 



Compound Interest 

Compound interest was commonly called Jewish interest, 
or profit, as is shown by the following from Van der Schuere : 1 

" When one wishes to gain money more quickly than can 
be gained in the usual time, then one must learn to reckon well 
what is his just due according to- the Jewish profit." 

The same tendency to associate compound interest with 
Jewish practice is seen in Riese : 2 

"A Jew lent a man 20 florins for 4 years, every half-year 
he added the interest to the principal. Now I ask, how much 
will the 20 florins amount to in 4 years, if every week the 
interest on 1 floren is 2 denarii? Answer, 69 florins 15 gr. 

n 21 256 480 2 804 5 . » 
9 TTST9 TOT & 9 16T 0. 

Equation of Payments 

This topic was. not commonly given. A few authors gave 

it separate treatment, but most of them condensed it into a 

few problems and placed thern under other rules, as that of 

interest. Trenchant states the object of the process thus: To 

1 Van der Scheure, Arithmetica, Oft Reken=const/ (1600 ed.). 

"Soo yemandt van t'ghevvin oock vvinst vvil heben snel, 
Als vvinste niet ibetaelt en vvordt ter rechter tijdt, 
So moet by leer en hier berekenen seer vvel, 
Wat hem met recbt toecomt, al ist een Ioodtsch profijt." 

Fol. Q 8 recto. 

2 Riese, Rechnung auff der Linien und Federn/ (1571 ed.). "Item/ ein 
Jude leihet einem 20 floren 4 Jar/ vnd alle halbe Jar rechent er den gewin 
zum hauptgut/ Nu frage ich/ wie viel die 20 floren angezeigte vier Jar 
bringen mugen/ To alle wochen 2jvon einem H gegeben werden? Facit — 
gewin vnd gewins/ t 2 69 floren/ 14 gr/ 9 6/ vnd Jifse 48 o||045 teil." 
Fol. Gv verso. 



l 4 6 SIXTEENTH CENTURY ARITHMETIC 

reduce to a single payment at one time several items payable 
at different times. 1 

Exchange and Banking 

Exchange as a business custom existed among the Greeks, 
from whom it was communicated to the Romans. It is known 
with certainty that Bills of Exchange existed about 309 B. C. 
and were introduced into Italy from Greece. In Rome private 
bankers were known as "Argentarii," and the practice of Ex- 
change as " Permutatio." 2 The subject as it has appeared in 
arithmetics was developed by the Italians in the sixteenth cen- 
tury. According to Unger(Die Methodik, p. 90) the earliest 
appearance of a bill of exchange was in Borgo's (Paciuolo's) 
Summa (1494), fol. 167. The earliest Italian bank was 
a kind of subtreasury of the Mint and was located at Venice. 
Public banking took its rise in that city in 1587. 3 Tartaglia 
gave four kinds of Exchange 4 and explained the conditions 
for acceptance, protestation, and return. The problems of ex- 
change are chiefly concerned with the translation of money 
units, weights and other denominate number tables from one 
system to another, as shown in Adam Riese : 5 

" 894 Hungarian florins are equal to how many Rhenish 
florins, when 100 Hungarian florins equal 129 Rhenish? Ans. 
11 53 ^- 5 P 2 i heller. Proceed thus: Add the exchange 
to 100 Rhenish and say that 100 Hungarian florins make 129 

1 Trenchant, Ari'thmetique (1578 ed.). " Remettre a vn iour de payment 
vne ou plufieurs parties payables a diuers termes." Page 316. 

2 See " Exchange, Roman," in Harper's Diet. Classical Lit. and Aritiq. 
(N. Y., 1897), 2:1597. 

3 €. A. Conant, A History of Modern Banks of Issue (N. Y., 1896). 

4 Tartaglia, Tvtte l'Opera (1592 ed.), II, fol. 174 recto. 
•Cambio (Exchange). 

1 . Minuto = common, meant changing money from one system to another. 

2. Reale = chief, meant expressing the value of a sum of money in dif- 
ferent places and covered remittances. 

3. Secco = dry, treated o<f drafts drawn on the maker. 

4. Fittitio = special kind of secco, meant bills drawn with various de- 
vices to prevent fraud. 

5 Riese, Rechnung auff der Linien und Federn/ (1571 ed.). "Item/ 894 
Vngerifch floren/ wie viel machen die Reinifch/ 29 auff? Facit 1153 
Reinifch/ 5 jS 2 heller/ und § teil. Thue jm alio/ Addir den Auffwechffel 
zu 100 Reinifch/ und fprich/ 100 Vngerifch thun 129 Reinifch/ wie vie! 
894 Vngerifch? Facit wie oben." Fol. F verso. 



THE ESSENTIAL FEATURES 



147 



Rhenish. How many Rhenish florins will 894 Hungarian 
florins make? Ans. Same as above." 

The questions often included the matter of remittances also. 
Thus, a person in Paris wishes to order the payment of 1200 
crowns in Augsburg; how many florins must be paid in 
Augsburg ? 

Since in the sixteenth century nearly every principality had 
its own mint and its own system of coinage, a treatment of 
exchange required a statement of the equivalents of many 
systems. Thus, Tartaglia treats of exchange between these 
places : 





Rome 




Venice 




Naples 




Pisa 




Lyons 




Siena 




Antwerp 




Naples 




London 




Bologna 




Paris 




Milan 


Venice 


Milan 




Barcelona 


and 


Pisa 




Provence 




Perugia 




Aquila 




Bologna 


Florence 


Sicily 




Genoa 


and 


Perugia 




Florence 




Rome 




Valencia 




Geta 




Palermo 




Avignon 
London 




Barcelona 




Genoa 


Avignon 


Paris 




Flanders 


and 


Florence 




Majolica 
Apulia 




Venice 




Rhodes 


Pisa 


Perugia 




Constantinople 


and 


Rome 








Barcelona 




Venice 
Genoa 




Venice 


Milan 


Avignon 




Milan 


and 


Pisa 




Genoa 




Paris 




Paris 






Bologna 


Pisa 




Venice 


and 


Rome 




Pisa 




Perugia 


Genoa 


Rome 




Ferrara 


and 


Palermo 




Siena 




Barcelona 
Paris 


Paris 


Bruges 






and 


Pisa 







148 



SIXTEENTH CENTURY ARITHMETIC 



It is easy to note from the problems of exchange the various 
articles of trade. A few are: saffron, wax, wool, soap, tin, 
sable, tallow, pepper, skins, furs, grain, ginger, cloves, camlet, 
caps, fustian, tapestry, taffeta, worsteds, musk, linen, satin, 
velvet, lead, iron, steel. 

The following is a partial list of articles which were items 
of exchange between the cities named : * 



Place sent from. 


Article. 


Place sent to. 


Breslau 


garments 


Vienna 


Bohemia 


wool 


Breslau 


Prague 


cloth 


Ofen 


Venice 


cloves 


Nuremberg 


Eger 


tin 


Nuremberg 


Venice 


saffron 


Nuremberg 


Nuremberg 


pepper 


Vienna 


Nuremberg 


pepper 


Breslau 


Basel 


paper 


Nuremberg 


Breslau 


wax 


Nuremberg 


Posen 


wax 


Nuremberg 


Augsburg 


almonds 


Vienna 


Nuremberg 


tin 


Augsburg 


Venice 


soap 


Augsburg 


Venice 


cloves 


Vienna 


Vienna 


wine 


Nuremberg 


Augsburg 


silver 


Vienna 



No adequate idea of the subject of Exchange can be given 
in a brief general article, for when the equivalents for weights, 
measures, and moneys of different systems are considered the 
matter increases to volumes. For a work of two hundred 
pages on this subject see Pasi, page 83 of this monograph. 

Chain Rule 

Although the name of this rule had a curious origin, 2 the 

1 Rudolff, Kunstliche rechnung mit der Ziffer und mit den zalpfennige/ 
(1534 ed.). Under Chapter on Wechfel. 

See also Exempel Buchlin (1530 ed.), fol. -d 8 recto. 

2 According to Cantor this term has its origin thus : In Menelaus's pro- 
position in which a line divides the sides of a triangle into six segments, 
the transversal was called sector (cutter). The Arabs translated this 



THE ESSENTIAL FEATURES I49 

meaning given to it in the sixteenth century was peculiarly 
appropriate. In the Arithmetics of that period it meant a rule 
to find the relation between two* denominate numbers meas- 
ured in different units by means of a series of intermediate de- 
nominate numbers. The following example from Riese will 
illustrate: 7 pounds at Padua make 5 pounds at Venice, 10 lb. 
at Venice make 6 lb. at Nuremberg, and 100 lb. at Nuremberg 
make 73 lb. at Cologne; how many pounds at Cologne do 
1000 lb. at Padua make? 

The work is arranged thus : 

7 lb. Padua = 5 lb. Venice 

10 lb. Vendee =6 " Nuremberg 1000 Padua 

100 lb. Nuremberg =73" Cologne 
Then, 7,000 lb. Padua = 2190 lb. Cologne, multiplying in 
columns. Therefore, 1000 lb. Padua=iooo lb. X fori = 312? 
lb. Cologne. 

The Italian plan of arranging the work brought out more 
clearly the significance of the name, Chain Rule. By their 
method the above problem would be solved thus : 




-Z&^SUtf O°o*><?~u«*.) 



Divide the product of the numbers on the broken line from 
Padua to Padua by the product of the numbers on the line 
from Cologne to Venice. Then 1000 lb. Padua= u^y.|£*Ai 
lb. Cologne = 312! lb. 

Examples of this nature were given by Brahmagupta x 
(c. 700 A. D.) and by Leonardo of Pisa 2 (1202). In Ger- 
many they were usually solved until 1550 by repeated ap- 
plication of the Rule of Three. Widman, 3 however, gave the 

al-katta, which appeared in the Latin of Leonardo of Pisa's Liber Abaci, 
as figua cata. Cantor, Geschichte der Mathematik, 2: 15. 

1 Unger, Die Methodik der praktischen Arithmetik, p. 91. 

2 Scritti di Leonardo Pisano, I, pp. 126, 127. 

8 Widman, Behede Rechnung (1489 ed.), fol. 152. 



i5o 



SIXTEENTH CENTURY ARITHMETIC 



Italian form, Apianus (1527) explained the difference be- 
tween the Chain Rule and the Rule of Three, Rudolff ( 1 540) 
gives some examples, and Stifel (1544) made a clear and 
formal explanation. The method reached England in the 
seventeenth century, for it appears in Wingate's Arithmetic 1 
(1668). 

The rule has several names, " Vom Wechsel," because of 
its connection with exchange, " Vergleichung von Mass und 
Gewicht," "Verweehselung von Mass und Gewicht," 2 " figua 
cata," 3 " regula del chatain," 4 " Regula pagamenti, 5 and 
" Kettensatz," the name that became general in Germany in 
the eighteenth century. 

Barter 6 
Although Barter was an extensive custom among primitive 
peoples, 7 it may seem strange that it should find place as a 
subject of instruction up to the last century. 8 There are 
two reasons why the subject was of sufficient importance in 
the sixteenth century to have given rise to a chapter in the 
arithmetics. 9 First, the scarcity of coined money, 10 and second, 
the custom of holding interstate fairs. 11 Not until the dis- 
covery of large quantities of gold in the New World was 
there a suitable metal in sufficient quantities to supply the de- 
mands of trade; hence, the direct exchange of goods was 

1 Villicus, Geschichte der Rechenkunst, p. 101. 

2 Widinan. 

8 Leonardo of Pisa. 

4 Ghaligai, Practica D'Arithmetica. 

6 W id man. 

6 Often called Stich Rechnung. Heer, Compendium Arithmeticae (1617 
ed.). Fol. Giij recto. 

7 W. Cunningham, The Growth of English Industry and Commerce (Lon- 
don, 1896), p. 114. 

8 It persisted in holding a place in the Arithmetics of the nineteenth 
century. See Pike's Arithmetic, 8th ed. (N. Y., 1816), p. 221. 

e Ciacchi, Regole generali d'abbaco (Florence, 1675), p. 114. 

10 See "Barter," New International Encyclopedia (New York, 1901-4). 
Cunningham, Cambridge Modern History, I, Chap. XV (London, 1902). 

11 Cataneo, Le Pratiche (1567 ed.), fol. 49 verso. 



THE ESSENTIAL FEATURES 



151 



essential to commercial progress. The great fairs which cor- 
responded to the International Expositions of the present time 
served to encourage this form of trade. Thus, the many 
technical questions about the expressions o'f values of goods in 
different systems and the methods of calculating the amount 
of one product to be exchanged for another necessitated a 
treatment of the subject in the arithmetics of that time. In 
barter the prices o<f articles were usually placed higher than in 
selling for cash. An example in barter from Baker reads : l 
1 Two marchants will change their marcadise, the one with 
the other. The one of them hath cloth of 7 s 1 d. the yard 
to sell for readye money, but in barter he will sell it for 8 s 4 cl. 
The other hath Sinamon of 4 s 7 d' the li. to sell for readye 
moneye. I demaunde how he shall sell it in barter that he be 
no loser." 

Alligation 

In the sixteenth century alligation found application chiefly 
in problems of the mint. Among others Rudolff gave these 
two problems : 2 

"A man has refined silver containing 14J lot per marck and 
coins containing 4^ lot per marck. How much of each will he 
need to make 40 marcks in which each marck will be 9 lot 
fine? Ans. 18 m of silver and 22 m of coins." 

"A mint-master has some refined silver containing 14^ lot 
per marck. How much silver and how much copper must he 
take in order to have 45 m, each marck being 9 lot fine? Ans. 
Pure silver, 27 m 14 lot 3 qn 2 ^V 9 ; copper, 17ml lot o qfi 

The following is from Thierfeldern : 8 

1 Baker, The Well Spring of Sciences (1580 ed.), fol. Wv verso. 

2 Rudolff, Kiinstlichc rechnung mit dcr Ziffer und mit den zalpfennige/ 
(1534 ed.). 

Similar problems are found in Rudolffs Exempel Buchlein (1530), fol. 
F g verso and G recto. 

8 Thierfeldern, Arithmetica (1587 ed.). "Item/ cin Herr hat dreyerlcy 
Gold/ wegen/ das erste 15 marck/ helt ein marck 15 karat/ 3 gran/ das 
ander 21 marck/ helt die marck 17 karat 2 gran/ das dritte 48 marck 



1 52 SIXTEENTH CENTURY ARITHMETIC 

"A man had three qualities of gold, the first contained 15 
marcks, each marck containing 15 karats 3 grains, the second 
contained 21 marcks, each marck containing 17 karats 2 
grains, the third contained 48 marcks, each marck containing 
12 karats 1 grain. What is the greatest weight the metal 
resulting from a mixture of these can have so that each marck 
may contain 14 karats 3 grains? Ans. The resulting metal 
will weigh 56^ marcks, for which he takes the first two and 
from, the third takes 29 jV marcks." 

Regula Fusti 

The Regula Fusti is an application of the Rule of Three to 
problems involving a reduction for impure or damaged goods. 
The problems refer to such commodities as spices, gold, silver, 
honey, and oil. Thus : 1 

"A sack of pepper weighs 3 centners 50 lb. The tare for 
the sack is 3! lb., each centner contains 11 lb. of fusti. One 
pound of fusti cost 4 gr. and a centner of pure pepper 72J fl. 
The question is: What is the pepper worth?" 

Another example is fromi Simon Jacob : 2 

"A merchant bought at Frankfurt a sack of cloves weighing 

2 centners 45! lb. The tare was 8i lb., and each centner con- 
tained 16 pounds of fusti. A pound of pure cloves cost 21 /?, 
and a pound of fusti 6 /?. He then went to Nuremberg. His 
expenses were 5i fl. For how much must he sell the cloves 

■helt 1 marck 12 karat/ 1 gran/ wie vil mag er von difen am meyften 
■befchicken/ das ein marck hake 14 karat/ 3 gran ? facit/ das Werck wird 
56^ ms. dar zu nimpt er die erften zwey/ vnd von dritten 29 J^ marck." 
Page 193. 

1 Thierfeldern, Arithmetiica (1587 ed.). "Item/ ein Sack Pfeffer <wigt 

3 cr. 50 tb. Thara fur den Sack/ 3 J lib, helt der cr. 11 lb. Fufti/ kost 
1 tb. Fufti 4 gr. vnd ein cr. lauter 72I fl. lit die Frag/ was der Pfeffer 
gefteihe? Fa. 231 U. o gr. 8 5 oi-g- hr." Page 119. 

2 Jacob, Reohenbuch auf den Linden und mit Ziffern (1599 ed.). "Item/ 
einer kaufft zu Franckfurt einen Sack mit Naglin/ der wigt 2 centner 45J 
lb. tara 8J pfundt/ helt der centner 16 pfund Fusti/ das lb. lauter vmb 
21 13. das lb. Fusti vmb 6. £. die bringet er gehn Nurnberg/ gestehen mit 
vnkosten dahin 5-J fl. wie soil er da selbst 1 lb. durch einander verkauffen/ 
das er vber alien kosten 30 gulden gewine. Vnd ich setze das Frankforter 
gewicht gleich dem Nurnberger ?" Fol. Mv verso. 






THE ESSENTIAL FEATURES 



153 



a pound that he may gain 30 guldens above all costs ? And I 
reckon the Frankfurt weight equal to that of Nuremberg." 

Virgin's Rule, also called Rule of Drinks. 

This rule (Regula, Virginum, or Regula Cecis) 1 grew out 
of the custom of charging men, women and maidens different 
prices for their drinks. Riese explains it thus : 2 "At times it 
chances that many people of different kinds are included in one 
bill and the reckoning is obscure as when men, women, and 
maidens are included in a reckoning for money spent in drink- 
ing and they are not to pay equally. To make such a reckon- 
ing you must study industriously this excellent rule, called the 
Rule of Drinks." 

Thierf eldern gives this example : 3 

" 47 people, men, women, and maidens together spent 47 gr., 
each man gave 5 gr., each woman 3 gr., and each maiden 1 hr. 
How many persons of each kind were there? Ans. 3 men, 4 
women, and 40 maidens." 

Rule of False Position (Single; Double) 

The Rule of False Position (Regula Falsi), essentially an 

algebraic process, is as old as Egyptian mathematics. It was 

used to solve various indeterminate problems. 4 Gemma 

Frisius explains the name thus : 5 " This rule which we are 

1 This name is derived from the Arabic ciintu Sekes, according to Zeu- 
then in LTnterm., 1896, p. 152 (quoted by Enestrom B. M., 10(2) : 96). 

2 Riese, Rechnung auff der Linien und Federn/ (1571 ed.). " lis begeben 
fich zu zeiten viel und maneherley rede unter den Leyen/ und unverften- 
-digen der Rechnung/ Als wenn Menner/ Frawen/ und Jungfrawen in 
einer Zeche verfamlet/ ein anzal gelds vertrincken/ und nicht zu gleich 
ibezahlen/ Solches zu machen/ foltu mit fleis diefe hiibfche Regel mercken/ 
welehe Cecis genant wird." Fol. Lvii recto. 

3 Thierfeldern, Arithmetica Oder Rechenbuch (1587 ed.). "Item/ 47 
Perfonen/ Mann/ Frawen und Jungfrawen/ haben verzehrt 47 gr. ein Mann 
gibt 5 gr. ein Fraw 3 gr. ein Jungfraw 1 hr. Wie vil find jeder Perfon 
in fonderheit. facit/ 3 Man/ 4 Frauwen/ vnd 40 Jungfrawen." Page 215. 

4 The Arabs called it the operation with scales, because of the figure, 
ZZXZZ, used in the method. Steinschneider, Abhandlungen, 3: 120. 

5 Gemma Frisius, Arithmeticae Practicae Methodus Facilis (1575 ed.). 
" Vocatur autem regula quam iam docemus, Falsi, non quod falsum doceat, 
sed ex falso verum elicere, fit q$ in hunc modum." Fol. F 3 recto. 



I 5 4 SIXTEENTH CENTURY ARITHMETIC 

now teaching is called the Regula Falsi, not because it teaches 
what is false, but because it teaches to find the true through 
the false." 

Tonstall * says of the name that the Arabs and Phoenicians, 
celebrated merchants, from whom arithmetic is thought to 
have originated, called this method of finding the truth by the 
foreign word, cathaym. The Latin races called it either the 
Rule of False Position or the Rule of False Assumption. 

Variations of this word are Kataim used by Cardan 2 and 
Helcataym used by Tartaglia. 3 Baker 4 speaks of this rule 
as the " Rule of Falsehoode, or false positions." 

Under applications of this rule Widman gives many number 
puzzles, as : 5 " You are to find for me a number to which if 
I add f of itself and divide the result by 4^, the answer will 
be 12." 

"Divide for me 15 into two unequal parts so that if I 
divide the larger by the smaller, the result will be 19." 

The following example from Onofrio will illustrate the 
method of working : 6 " This principle is illustrated by several 

1 Tonstall, De Arte Supputandi (1522 ed.). "Arabes et Phoenices mer- 
catura celebres, et a quibus Arithmetica profecta primum putatur: artem 
illam ueritatis inueniende barbaro uocabulo Cathaym appelant. Latini fiue 
falfaru pofitionu, fiue falfaru c5iecturaru regulas uocat. Fol. r 3 recto. 

2 Cardan, Practica Arithmetica (1539 ed.), Chap. 47, Lii verso. 

3 Tartaglia, La Prima Parte Del General Trattato (1556 ed.), Book 16. 

4 Baker, The Well Spring of Sciences (1580 ed.), fol. Zv verso. 

5 Widman, Behend und hupsch Rechnung (1508 ed.). " Du solt mir 
suche ein zal wen ich -J der selben zal dar zui addir/ vn darnach das aggre- 
gat in 4-I partir/ das mir 12 kumen." Fol. e 3 recto. 

" Diuidir mir 15 in 2 teil die vngleich sein/ vnd we ich dz grost diuidir 
durch dz kleinst das 19 kume/. 

G Onofrio, Arithmetica (1670 ed.). "E per dar principio a gl' effempij 
fia quefto il primo. Ottauiano Semproni eompro tre diamanti ; il fecondo 
li cofto on. 4. piu del primo, & il terzo quanto il primo, e fecondo, & on. 
5 piu; in tutto fpefe on. 81. quanto dunque li cofto ciafcun diamante? 
Per folutione della prefente domanda, fupponi il primo diamante efferli 
coftato on. 24. il fecondo, perche dice la domanda, che li cofto on. 4. piu 
del primo; li fara coftato on. 28. & il terzo, perche cofto quanto il primo, 
e fecondo, & on. 5. piu, dunque cofto on. 57. la fomma delli quali tre 
numeri s'e on. 109. & eglino coftarono on. 81. dunque la noftra pofitione 



THE ESSENTIAL FEATURES I55 

examples, of which this is the first : Ottaviano Semproni 
bought three jewels, the second of which cost 4 on. more 
than the first, the third cost 5 on. more than the first, and the 
third cost 5 on. more than the first and second together, and 
all three cost 81 on. Required the cost of each. In order to 
solve, suppose for the present that the first jewel cost 24 on., 
the second, because it was to cost 4 on. more, cost 28 on., the 
third, because it was to cost as much as the first and second 
together and 5 more, cost 57 on. The sum of these make 
109, and since they are to cost 81 on., then our position is 
false by excess; this excess (since 28 is the result of taking 
81 from 109) will be designated by the letter, P, in this man- 
ner 24 P. 28. 

fu falfa per ecceffo, quale ecceffo (che fono on. 28. perche tanto auanza 
il nnmero 109. al numero 81.) fi notera con la lettera P, in quelta maniera. 
24. P. 28. 

" Faccifi vn' altra nuoua pofitione e fuppongafi il primo diamante hauer 
coftato on. 20. il fecondo, perche cofto onze 4. piu, fara conftato on. 24. 
& il terzo, perche cofto quanto il primo, e fecondo, e 5 piu, hauera coftato 
on. 49. fommati quefti tre numeri fanno on. 93. & eglino 
doueano fare on. 81. dunque habbiamo di nuouo auanzato dalla 24. P. 28 
verita per on. 12. e pero noteremo quest' errore parimente con 20. P. 12 
la lettera P, cosi dunque ftara l'effempio. 

" Hor per trouare la verita mediante la proportionality della pofitioni 
con quella degl' errori, cosi s'operira. Perche l'vna, e l'altra pofitioni haue 
auanzato la verita, fi fottrarra il minore errore dal maggiore, cioe 12. da 
28. e rimarra 16. quale fi notera fotto per partitore : doppo fi moltiplice in 
croce la prima pofitione, cioe 24. per il fecondo errore 12. & il prodotto 
288. fi fcriuera alia parte deftra del medefimo errore 12. come in queft' 
eff empio appare : parimente fi moltiplichera la feconda pofitione 20. per 
il primo 24. P. 28 560 

>< 

20. P. 12 288 



Partitore 16. 272 Partitione 

Quotiente 17. 112 

— o 
errore 28. & il prodctto 560. fi fcriuera dalla parte deftra del medefimo 
errore 28. delli quali due prodotti fottratto il minore dal maggiore, cioe 
288. da 560. reftera 272. da partirfi al partitore 16. fi che partendo 272. a 
16. il quotiente fara 17. & onze 17. cofto il primo diamente, il fecondo on. 
21. cioe on. 4. piu, che il primo, & il terzo on. 43. cioe quanto il primo, e 
fecondo, e 5 piu, quali tre numeri infieme vniti fanno on. 81. come nella 
domanda fi cercaua." Fol. Ee„ verso. 



156 SIXTEENTH CENTURY ARITHMETIC 

" Take a new position and suppose that the first cost 20 
on,, the second, since it cost 4 on. more, would cost 24, and 
the third, because it was to cost 5 more than the first and 
second together, would cost 49 on. The sum of these would 
then be 93, where it should be 81, then we have 
the new variation from the truth, 12, which error 20 p ' a 

we designate by the letter, P, as in the example. 

" The truth is found by finding the mean proportionals be- 
tween these positions together with their errors. 

" Since the former and the latter positions vary from the 
truth, if the smaller group is subtracted from the larger, as 
12 from 28, there remains 16, which we place below for a 
divisor. Then we multiply crosswise the first position, 24, 
by the second error, 12, which gives the product, 288, which 
we place at the right of the error, 12, as shown in this example : 

24. P. 28 560 

>< 

20. P. 12 288 



Partitore 16. 272 Partitione 

Quotiente 17. 112 

— 

" Then multiply the second assumption, 20, by the first 
error, 28, and the product is 560, which is placed at the 
right of the error, 28, then the difference between these pro- 
ducts is found which is 272. When this remainder is divided 
by 16, the quotient will be 17, therefore the first jewel cost 
17 on., the second 21,4 on. more than the first, and the third, 
43, equal to 5 more than the first and second; the three to- 
gether make 81, as the problem required." 

Most of the minor rules, mentioned p. 132, are special cases 
of other rules. That is, they designate more minute divisions 
used by a few authors for particular problems and included by 
most writers under more general rules. Thus, voyage 1 was 
commonly used to stand for courier problems. 

Problems of the mint were very important at that time, be- 

1 Van der Schuere, Arithmetica, Oft Reken = const (1600 ed.), fol. 
Ziiii recto, gives the Hound and Hare problem, the Mule problem, and 
other courier problems. 



THE ESSENTIAL FEATURES j^y 

cause the coinage of money was delegated to local authorities 
and on account of the multiplicity of standards. Although 
these questions were often; treated under Alligation, many 
authors grouped them: under the title Mintage. 1 

Certain practical arithmetics emphasized solutions of ques- 
tions of householders and landlords and designated the prob- 
lems by such titles as Salaries of Servants and Rents. 2 

Among the many safeguards which European nations have 
thrown about general public interests for centuries is the legal 
standardizing of bread. The weight of a loaf of bread which 
sold for a fixed price was regulated according to> the price of 
the grain from which it was made. The earliest regulation 
yet found is the Frankfurt Capitulare (794 A. D.). Londoo 
regulations are found as early as the twelfth century. An- 
other good specimen is the "Assize of Bread " of the time of 
Henry II. The general law which was practically followed 
was : The weight of the loaf varies inversely as the price of 
wheat. 3 The following from Finaeus is a typical bread prob- 
lem, as found in the arithmetics of the sixteenth century : 4 
"When a bushel of wheat is sold for 34 shillings (for ex- 
ample), and the bread made from it is sold at 6 denarii per 
loaf, one observes that the weight is 12 ounces; if the same 
bushel of wheat is sold at 28 shillings, how many ounces must 
be put into each loaf to sell for 6 denarii ?" 
Overland Reckoning 

The title, Rechnung Uber Land, really a synonym for Ex- 

^-Van der Schuere, Arithmetica Oft Reken = const (1600 ed.), fol. Yiiii 
recto. 

2 Unicorn, De L' Arithmetica universali (1558 ed.), fol. Ccccc^ verso. 

3 W. Cunningham, The Growth of Industry and Commerce during the 
Early Middle Ages (London, 1896), p. 68. 

An excellent explanation of the English Law in 1800 is found in Nasmith, 
An Examination of the Statutes now in Force Relating to the Assize of 
Bread (Wisbech, 1800). 

4 Finaeus, De Arithmetica Practica (1555 ed.). "Cum medimnus tritici, 
uaenit (exempli gratia) duodenis 34, & confectus ex illo panis 6 denarioru 
turonen ohferuat podus 12. unciarum: fi idem medimnus tritici, uenerit 
ad pretium 28 duodenorum, queritur quot unciaru formandus erit idem 
panis 6 denorioru?" Fol. Sij recto. 



1 58 SIXTEENTH CENTURY ARITHMETIC 

change, was used by some writers in a broader sense, namely, 
to include problems concerning the purchase of foreign goods 
as well as the methods of remitting money. Trenchant x 
gave twenty-three pages to the subject as well as several ap- 
pendices and included in it the treatment of Exchange, explain- 
ing four cases similar to those of Tartaglia. See page 146, 
note 4 of this monograph. 

Because problems on gain and loss, interest and discount 
were abundant, one naturally seeks for a treatment of per- 
centage. But percentage as a separate subject did not appear 
until the end of the sixteenth century. There appeared, how- 
ever, under the various subjects, problems of that nature. The 
symbol, %, although not in the text-books of that period, 
originated about the beginning of the fifteenth century. 2 

The following are among the various expressions used for 

per cent : 

p cr. 3 von 100 4 pour 100 5 ten 100 c 

mt hundert 7 int hundert 8 met 100 ° .mit 100 10 

per cento 11 pro 100 12 p 100 13 uppon the 100 14 

an 100 15 p J 16 auff 100 17 per C. 18 

1 Trenchant, L'Arithmetique (1578 ed.), p. 340. 

2 The origin of the present sign, %, as an abbreviation for per cento 
has recently been traced by Dr. David Eugene Smith, Columbia Univer- 
sity, New York, to a manuscript of the first half of the fifteenth century. 

3 E. g., Heer, Compendium Arithmeticae (1617), fol. F recto ct al. 

4 E. g., Heer, fol. Fv verso. Jacob, Rechenbuch auf den Linien und mit 
Ziffern (1599 ed.), fol. Mv recto. 

5 E. g., Wencelaus, T'Fondament Van Arithmetica (1599 ed.), under In- 
terest. 

e Ibid, 

7 E. g., Van der Schuere, Arithmetica (1600 ed.), under Gain and Loss. 

8 Ibid. ° Ibid. 10 E. g., Jacob, under Gain and Loss. 

11 E. g., Chiarini and Jacob, under Gain and Loss. 

12 E. g., Finaeus, De Arithmetica Practica (1555 ed.), fol. Si verso. 

13 E. g., Unicorn, De L'Arithmetica vniuersali (1598 ed.), under Interest. 

14 E. g., Baker, The Well Spring of Sciences (1580 ed.), fol. Riiii verso. 

15 E. g., Heer, fol. Fiij recto. 

10 E. g., Giocomo Filippi Biordi, Arithmetica et Prattica (MS.) (1684). 
He uses " 12 p £ -f J" for 124%. 

17 Jacob, " Setz 93 fl. losung (verftehe aufi 100 fl.) gehen 3J fl. wie viel 
117 fl." Fol. Mv recto. 

18 Chiarini, Qvesta e ellibro che tracta de Mercatanti et vsage de paesi." 



THE ESSENTIAL FEATURES I59 

Pussies 

Besides the practical problems which were classified under 
the general rules, there were many problems famous as 
puzzles or amenities. The work of Bachet de Meziriac * 
(1624), a book now generally accessible through modern 
editors, is a collection of such problems known in his time. 
The following list is incomplete, but it contains the most in- 
teresting of those contained in the arithmetics consulted in 
preparing this monograph; the writers mentioned are those in 
whose arithmetics the problems appeared, but not the origin- 
ators of the problems. Only the essential feature of each 
problem is stated : 

Potato Race. One hundred stones (or potatoes) are placed 
in a row, the adjacent stones being 1 yard apart; how many 
yards will one have to travel in starting with the first and 
bringing each stone separately to the position of the first? 
(Trenchant). 

Snail in the Well. There is a well 20 fathoms deep. Every 
day a snail climbs 7 fathoms and at night falls back two 
fathoms. In how many days will he come from the well? 
Ans. 3f days. (Rudolf!, Riese, and others). 

Chess-board Problem. Required the number of kernels of 
wheat needed in order to place 1 kernel on the first square of 
a chess-board, 2 on the second, 4 on the third, and so on for 
the 64 squares. Given by Masudi, (Cairo, 950) in " Meadows 
of Gold." See Bone. Bull., 13:274. 

Eating and Drinking Problems. Some hunters with 
loaves of bread and bottles of wine meet at a spring; they 
seek to divide the refreshments so that each shall share ac- 
cording to what he brought. (Ghaligai, fol. 66 recto). 

Similar problems about drinking wine were given by Re- 
corde. In the Dutch arithmetics this problem takes the form 
of a dispute between a lion, a wolf, and a dog over their prey. 

Horseshoe Problem. A man agrees to pay one penny for 
the first nail, 2 pence for the second, 4 for the third and so on 

1 Claude Gaspard Bachet de Mezeriac, Problemes plaisants (1624). 



160 SIXTEENTH CENTURY ARITHMETIC 

for all the nails used in shoeing his horse; how much does 
he pay? This is similar to the Chessboard problem. 

Courier Problems. The name is now used to represent the 
whole class of problems that concern the movement of bodies 
at given rates in which some position of these bodies is given 
and the time required before they will assume another given 
position. The name originated with the French to designate 
problems about messengers delivering despatches in connection 
with military service. 

The problems of the clock hands and the times of conjunc- 
tion of the planets fall into this general class. 

Courier problems appear in the Bamberg Arithmetic 
(1483) under the title, "Van Wandern." They are found 
in Calandri, Ton-stall, Kobel, Cardan, and Trenchant. 

Three Casks. Three casks together con- 
tain 79 gal.; the second contains 3 gallons 
more than i as much as the first, and the 
third contains 7 gallons less than the 
second ; how many gallons are there in each ? three jugs. 

(Trenchant, Baker, Kobel.) 

God Greet You Problem. God greet you with your 100 
scholars! We are not 100 scholars; but our number and the 
number again and its half and its fourth are 100; 'how 
many are we? (Rabbi Ben Ezra, Alcuin, Leonardo, Gram- 
mateus, Riese, Kobel and Van der Schuere.) 

Thief Problem. A thief having robbed a castle met a guard 
in trying to escape whom he bribed with i of his plunder; 
at the next gate he met a guard whom he bribed with i of 
what he has left ; he escaped with 1 5 lb. How much did the 
owner of the castle lose? (Tonstall and Kobel.) 

This problem ran through many variations as the plunder- 
ing of gardens and the stealing O'f apples. 

Reed Problem. A reed standing in the center of a circu- 
lar pond 12 feet deep projects 3 ft. out of water; when the 
wind blows it over to the side of the pond, it just reaches the 
surface; what is the distance across the pond? 

1 This illustration is from a fourteenth century manuscript. 





THE ESSENTIAL FEATURES j6i 

Tree Problem. A tree 50 ft. high was broken so that its 
top touched the ground 30 ft. away from its base. How much 
was broken off and how much remained standing? (Calandri) . 

Mill-Wheel Problem. A mill has 5 wheels, the first wheel 
grinds 7 staria of wheat in 1 hour, the second 5 staria, the 
third 3 staria, the fourth 2, and the other 1. In how many 
hours will they altogether grind 50 staria? 

Jealous Husband Problem. A boatman 
has his wife, two strangers, and their 
wives to ferry across a stream. His boat 
will carry only two persons. Being jealous 
of his wife he is not willing to leave her boat and dock. 1 
with either of the strangers. How many trips must he make 
to ferry the party across and not leave his wife with a stranger? 

A familiar form of this problem is that of the boatman with 
a fox, a goose and some corn, or with a wolf, a goat and 
some cabbage. 

Cistern Problem. See page 134. Many variations of this 
problem are still found in text-books concerning the building 
of walls, digging of trenches, and the famous " If A can do a 
piece of work in 5 days, and B can do it in 8 days, how long 
will it take both working together to do it ?" 

Market Problem. A woman going to market with a basket 
of eggs found that when she counted them by twos there 
was one over, but when she counted them by threes there were 
two over. The whole number was between 50 and 60; how 
many were there in the basket? (Baker.) 

Mule and Ass Problem. A mule asked an ass whether 
his load was heavy ; the ass replied : " My load is thrice as 
heavy as yours, but, if I had yours and mine together, it would 
be only half a ton; find the result yourself." (Gemma 
Frisius.) 

Servant Problem. A master bargained with a servant to 
give him 10 guldens a year and a coat. The servant re- 
mained only 7 months. At that time the master said : " Leave 
my house and take the coat that I gave you ; I owe you noth- 

1 This illustration is from a fourteenth century manuscript. 



162 SIXTEENTH CENTURY ARITHMETIC 

ing more. How many guldens was the coat worth ?" (Uni- 
corn and Kobel.) 

Casket Problem. A jewel casket and lid weigh 27 oz. ; the 
lid is f as heavy as the casket; what is the weight of the lid?" 

Striking of Clock. Venetian clocks strike from 1 to 24; 
how many days and nights go by for 300 strokes of the clock? 

Tower Problem. One-third of a tower is hidden under 
the earth, a fourth is submerged under water. 60 cubits rise 
above the water. It is desired to know how many cubits are 
under the earth and how many are submerged under water. 
(Tonstall.) 

Garrison Problem. A captain with 4000 soldiers was be- 
sieged in a fortress by the enemy for 7 months. They had 
only provisions for 5 months and were without hope of re- 
ceiving any during the siege of 7 months. It is required to 
know how the captain shall apportion the rations that they 
may last during the siege. (Champenois.) 

Will Problem. A man on his death-bed made his will thus : 
If his wife (about to be confined) should bear a son, he 
should receive i of the property valued at 3600 aurei, if she 
should bear a daughter, the daughter should receive a third. 
She gave birth to a son and a daughter; it is required to know 
how much each should receive. This question was also known 
as the Widow Problem and the Problem of Inheritance and 
dates back to 1 the Greeks and Romans. It was stated by 
Salvianus Julianus in the reign of Hadrian. (Given by 
Gemma Frisius, Tartaglia, Rudolff, Trenchant, Widman.) 

Hiero's Crown. Vitruvius relates in Book 9, Chapter 3, 
that when Hiero, the king, had decided to make an offering 
to the gods of a crown of pure gold, he entrusted it to a work- 
man, who (as they are always wont to do) mixed a portion 
of silver with the gold. A fraud was suspected when the 
crown was finished and Archimedes of Syracuse detected it 
thus: He obtained a mass of pure gold of the same weight 
as the finished crown and another mass of pure silver of the 
same weight. He placed each separately into a vessel filled 
with water saving the water which flowed out each time from 



THE ESSENTIAL FEATURES ^3 

the vessel and thus found the amount of gold and silver. 
Let us suppose the weight of the crown and the two pieces 
of metal to have been 5 lb. each; 3 lb. of water overflowed 
from the immersion of the gold, $i lb. from that of the crown, 
and 4§ from that of the silver. Therefore the question is : 
how much gold and how much silver were there in the crown? 
(Gemma Frisius.) 

Statue of Minerva Problem. I am the statue of Minerva; 
my gold, however, is the gift of the youthful poets. Charisius 
furnished J, Thespis -J; Solon, ^-, Themison, -^ ; the re- 
mainder, nine talents, was the gift of Aristodicus. (Ramus.) 

Herds of Alcides. To one asking the number in the herds 
of Alcides it was replied that i were near the gently flowing 
Alpheus, i grazed on the hill of Saturn, T V on the mountain 
of Tarixippus, -^V near the divine Elides, and -gV in Arcadia. 
The rest of the herd is 50. 

Beggar Problem. Three mendicants approached a priest 
holding a purse of money to be distributed among the poor. 
Having compassion for their poverty he gave to the first one- 
half of what he had in the purse and 2 nummi ; to the second 
he gave half of what was left and 3 nummi besides; to the 
third he gave 4 nummi more than half of what was left. Only 
one nummus remained in the purse. It is required to find 
how many nummi were in the purse at first. (Tonstall.) 

Ring problem. To find who has a ring in a company. If 
one person in a company standing in line has a ring on a cer- 
tain finger and you wish to know which one has it and on 
which finger it is, have one of the company silently double 
the number denoting the order of the one who has the ring, 
add 5, multiply this by 5, add the number of the finger on 
which the person has the ring, and tell the result. Take 
away 25 and the tens' digit will be the number of the person 
and the units' digit the number of the finger. (Trenchant.) 

Mensuration 
Besides the commercial applications, arithmetic of the six- 



164. SIXTEENTH CENTURY ARITHMETIC 

teenth century was very serviceable in the field of mensura- 
tion. Practical arithmetics commonly contained a section de- 
voted to mensuration. The combination of the Reckoning 
Book (Rechenbiich) and the Mensuration book (Visirbuch) 
by Kobel * represents the two forms of arithmetic at that time. 
The Visirbuch was better illustrated than any of its contem- 
poraries. The name, Visir, means gauge and Visirbuch, 
technically gauge book, means a book to teach gauging of 
casks. Its contents, however, were concerned with all forms 
of practical mensuration. 

The following list given by Cataneo 1 furnishes an idea of 
the extent of the subject of mensuration: 2 

The Measurement of Wood. 3 

The Measurement of Solid Bodies. 4 

The Measurement of Triangular Prisms. 5 

The Measurement of Square Prisms. 6 

The Measurement of Square Pyramids. 7 

The Measurement of Parallelepipeds. 8 

The Measurement of Walls. 9 

Another Method of Measuring Walls, Floors, Surface to 
be Whitewashed. 1 ° 

The Measurement o»f Casements. J1 

The Measurement of the Scarp of a Wall. 12 

1 Kobel, Zwey rechenbuchlin. " Uff der Linien vnd Zipher/ Mit eym 
angehenckten Visirbuch/ so verstendlich fur geben/ das iedem hieraufi on 
ciii lerer wol zulernen " (1531). 

2 Cataneo, Le Pratiche Delle Due Prime Matematiche (1567 ed.). 

3 Del Misvrar De I Boschi, fol. W 4 verso. 

* Del Qvadrar Le Cose Corporee, fol. Xi recto. 

6 Del Qvadrar Le Colonne Triangulari, fol. Xi recto. 

6 Del Qvadrar Le Colonne Qvadrangulari, fol. Xi verso. 

7 Del Qvadrar Le Piramide Qvadrangulari, fol. Xi verso. 

8 Del Riqvadrar I Vasi Qvadrangulari, fol. Xi verso. 

9 Del Riqvadrar Le Muraglie, fol. Xij recto. 

10 Altro Modo Di Riquadrar Muraglie, palchi, sciabli, e legnami, fol. Xij 
recto. 
11 Del Misvrar I Casamenti, fol. Xij verso. 
"Del Riqvadrar Le Scarpe De I muri, fol. Xij verso. 



THE ESSENTIAL FEATURES 1 65 

Further subjects are: 

The Measurement of Bodies with Square Surfaces. 

The Measurement of Floors and the Number of Bricks 
Necessary for a Square Surface. 

The Method of Finding the Number of Bricks in a Wall. 

The Measurement of Round Bodies, first the Measurement 
of a Ball. 

The Measurement of a Cistern. 

The Measurement of a Cylinder. 

The Measurement of a Cone. 

The Measurement of a Pile of Grain. 

Another Method of Finding the Contents of a Pile of Grain. 

A still different Method of Finding the Contents of a Pile 
of Grain. 

The Method of Finding the Capacity of a Cask. 

The Method of Finding the Capacity of a Barrel. 

Another Method of Finding the Capacity of a Barrel. 

The Method of Finding the Capacity of a Bin of Grain. 

The Measurement of a Small Cylinder. 

The Measurement of Cylindrical Walls. 

Besides the guage which served to measure the capacities of 
castes and small receptacles, another instrument, called the 
quadrans, or quadrant, was used to measure cisterns, walls, and 
distances. There were two forms of the latter instrument, one 
a square with graduated sides and the other a quarter of a 
circle with a graduated arc. Although the proportionality of 
the corresponding sides of similar triangles was the chief prin- 
ciple used in the solution of problems, the graduated arc in 
connection with the plumb-line made possible the measurement 
of angles. In this form of the quadrant we recognize the be- 
ginnings of the modern theodolite. 

The following page from Finaeus (1532) illustrates the 
method of rinding heights and distances by use of each form 
of the Quadrans. 



1 66 



SIXTEENTH CENTURY ARITHMETIC 



PINES 



D E t ? H. 



Exctnptom, 



Aliatcifckm 
cbfenmionlJ 




O R O N T I I 

20 primi clementorum Euclidis facile manifeftatur . & anguhis A B h ; an* 
uloAG F eft »qiialk(JiamutercprecTro) igitur per 4fcxtidufdcmEucL'dis, 

% ficut H B ad B A, ica F C putd laricudoad C A compoGtam ex G B & b a longi- 

tudmcm.fiue profunditatetm > 

Sit exempli gra'tia BH2o partium/pialnim latus quadrau' eft 6o:b e aute me* 

tiatur,&: fit in exemplum 6 cubitorum,tot etiam cubitorum erit G F:funt enim la 

teraperaUelogram*ndBEPGoppofita,qua:per*4 eiufdem primi funt inuicera 

alqualia.Duc igitur 6 in e»Q,fient $6b:qua: diuideper 20, 8C.hshebis pro quorie* 

te i8.Tot igitur cubitois erit A G: 

A quad dempfcris A B triura ucr* 

bi gratia cubitorum, relinqwtur 

B G dcfyderata Be in profundum 

depffa puceHogitudoiy cubitoRt. 

IDEM Q.V O Q_V E SIC OB* 

u'nebis. Metire H e: fitqj exempty 

caufa 5 cubitoru. Deindemulti* 

plica 5 per 6o,fient joo:hxc diui 

per iOjproducentur i^uelut an* 

tea.Bina nana/ triangula A B H et 

HEF funt rurfum scquiangul; 

quoniam angulus A H B angulo 

E H f ad uerticem pofito , per iy 

primi Euclidis eft a:qual{s.ite re* 

c*his qui ad b, re&oqui ad E pari 

ter acquat. rcliquus igitur BAH 

rcliquo H F E per JZ eiufdem pri- 
mi eft arqualis. Vndc per fupe* 

riusallegara quarts propofitione 

fcxti,ficut hc ad b A,ita H E ad E F.eidem B G,per hypothefim arqualem. 
Cum autem accident puteum rotundam habere figuram.habenda eric cofydc* 

ratio diametri putealis orificr},& fe liqua omnia ueludprfus abfoluenda. 

58)R.EIIQ.VVHETS,VT 4 

eandem rerum in profunda de* 

preffarum, per uulgatu quadra* 

tcm metiri doceamus altitudine. 

Sit itaqj puteus drcularis E F G 

H^cuius diameter fit E F,aut illi 

arqaali's G H* Adplica igitur qua* 

dratem ipfi putei orificio: in hue 

modest finis lateris A D ad datu 

pun&umEconftituatur. Leua 

poftmodu,autdeprime quadra* 



temClibero femper demiiTo per* 
pendiculo)donec radius uifualis. 
per ambo foramina pinnacidioRi 
ad inferiorem & e diametro Cu 
f ™ Xl * crmi ™ "perducat.Quo 
Jafto & immoto quadrate, uide 
in qua 








A typical problem of this kind is the following from 
Belli : x 

1 Belli, Silvio, Qvattro Libri Geometrici (1595 ed.). "La ragione 
e qvefta, l'angolo del triangolo AEB, enguale al l'angolo B del triangolo 
CEB, perche l'vno, e l'altro d'effi e retto, & l'angolo E e commune ad 
amendue i detti triangoli : Onde per la trigefima feconda del primo libro 
de gli elementi d'Euclide, il reftante angolo del l'altro. Et per la quarta 
del fefto i lati, che riguardano gli angoli vguali fono proportionali. Adun- 



THE ESSENTIAL FEATURES 



167 

Required to find the distance from point E to point A 
and the distance from B to A as shown in the figure. The ex- 




planation is as follows : The angle (B) of triangle AEB equals 
the angle B of the triangle CEB, because they are right angles; 
and the angle E is common to both of these triangles. Then 
by the thirty-second proposition of the First Book of Euclid's 
Elements the remaining angles are equal, and by the fourth 
proposition of the Sixth Book, the sides opposite to the equal 
angles are proportional. Then BC is to BA as EC is to EA, 
as the side BE of the small triangle is to the side BE of the 
large triangle. And the side BE of the small triangle, as was 
presupposed, occupies as many small divisions of the square as 
there are paces in the side BE of the large triangle, where- 
fore the small divisions of the side BC of the small triangle 
are as many as the paces of the side BA of the large triangle 
which was the first thing sought. And in the same way the 

que la proportione del lato BC al lato BA, & dello EC alio EA, fi come 
del lato BE del picciolo, al lato BE del grande, & il lato BE del picciolo, 
dal prefuppofito ha tante delle particelle del lato del Quadrato quante 
fono le paffa del lato BE del grande: per la qual cofa ancor le particelle 
del lato BC del picciolo fono quante le paffa del lato BA del grande, che 
e il primo intento. Et per lo medefimo modo le particelle del lato CE 
del picciolo triangolo fono vguali per numero alle paffa del lato AE del 
j grande, che e il fecondo." Fol. A 4 verso. 



j68 sixteenth century arithmetic 

... 

number of small divisions of the side CE of the small triangle 
is equal to the number of paces of the side AE of the large 
triangle, which was the second thing sought. 

Thus applied arithmetic represented the vital commercial 
interests of that period; it naturally contained some traditional 
and artificial problems, but in the main it expressed the quan- 
titative side of contemporary trade and industry. 

Summary 

The foregoing exposition of the subject matter of arith- 
metic belonging to the first century of printed books shows 
that this was an important period in perfecting the algorisms, 
or the processes of present day arithmetic with the Hindu nu- 
merals. From the treatment of processes (pp. 36-37) it ap- 
pears that the addition, subtraction, multiplication, and divis- 
ion of integers reached their highest development in the six- 
teenth century. Some forms of these processes have since 
been discarded, 1 but all that are in current use to-day are an 
inheritance from that period. All processes with common 
fractions now used, including some graphical methods, were 
practiced in the sixteenth century (pp. 85-110). Some meth- 
ods, however, were rare, as multiplying by the reciprocal of 
the divisor in division (p. 107). Arithmetic, geometric and 
harmonic progressions (pp. 1 10-1 16) as finite series, were com- 
monly explained by particular examples. The processes of in- 
volution, and evolution were practically complete (pp. 121-127). 
Even the use of tables to save calculation was a common prac- 
tice. (Tartaglia, p. 58; Trenchant, p. 144, and Jean, p. 82.) 
Thus the elementary processes reached their maturity except 
in the case of decimals and logarithms. The decimal frac- 
tion, the prefection of the processes with decimals, and 
logarithms were contributions of the seventeenth century. 

The constructive period of applied arithmetic began about 
1400 and lasted for more than a century. Mensuration 

1 For example, addition and subtraction in which the result was placed 
above the addends or above the minuend; multiplication by complements; 
multiplication by tables above 12 X 12; multiplication by fancy geometric 
forms ; division by galley and other methods. 



THE ESSENTIAL FEATURES jfy 

(p. 163) and commercial problems (p. 132) received exhaustive 
treatment at the hands of Paciuolo, Borgi, Tartaglia, Rudolff 
and Riese. Little has been added to the work of these writ- 
ers from that time to the present, except in such particulars as 
industrial problems, insurance, compound interest and ex- 
change. There are few problems found in present school 
arithmetics whose types were not foreshadowed in the 
arithmetic of that century. The sixteenth century problems 
in accounts, exchange, transportation, gain, loss, percentage, 
discount, salaries, rents, taxes, interest, tare, duties and part- 
nership testify to this fact (pp. 132-139). 

Even the methods of solution were more like those now in 
use than is generally supposed. Besides the solution of in- 
determinate problems, which were really solved by algebra in 
the form of rules as now, there were two methods of solving 
business problems: the method of unitary analysis (pp. 137- 
138) and the Rule of Three (pp. 132-139). 

The actual contributions to the theory of arithmetic made 
in the fifteenth and sixteenth centuries were few. The first 
great arithmetic printed in Europe, that of Paciuolo (1494), 
doubtless possessed some originality, but its chief merit con- 
sisted in its being a systematic and exhaustive summary of all 
the pure mathematics known at that time, and little was added 
to Paciuolo's arithmetic by way of theory in the sixteenth 
century. There were certain variations in processes, as the 
reduction of the number of species (pp. 36-37), different or- 
ders of presentation, and graphical illustrations (pp. 104-106) 
which may be called improvements, but the processes with in- 
tegers and common fractions were practically complete before 
the era of printed books. With symbolism the case was dif- 
ferent, for the improvements in methods of notation of whole 
numbers (pp. 37-39) and of decimal fractions (p. J2) and in 
the symbols of operation (p. 53) show that arithmetic was 
in a rapid state of development. Besides the advance in no- 
tation and symbolism, great progress was made in applied 
arithmetic; the cumbersome form of unitary analysis (as used 
by Calandri, p. 138) developed into the equational form seen 



I JO SIXTEENTH CENTURY ARITHMETIC 

in Thierfeldern ; the arbitrary Rule of Three came to be sim- 
ple proportion; exchange came to embrace a comparison of 
the denominate number systems of all Europe and the Levant ; 
and finally, many processes were superceded by tables of re- 
ference (pp. 82,, 144). 

The Place of Arithmetic in the Schools of that 

Period 

Arithmetic in the Latin Schools 
The Latin Schools were the chief agency through which 
the great reformers and teachers of the Renaissance recon- 
structed and extended elementary education. Seventy 
gymnasia * were founded in the kingdom of Prussia during 
the sixteenth century. These were not everywhere known, as 
in Germany, by the name of " Latin Schools," for in England 
they received the name, " Grammar Schools," and in France 
and Italy they were known as the " Schools of the Teaching 
Orders." 

The chief functions of these schools were to> teach the Latin 
language and to furnish general culture. A knowledge of 
Latin was necessary at that time for one who wished to be 
educated, for the standard works of literature, science, philo- 
sophy, law and theology were all expressed in the classical 
languages. 

The courses of instruction in these schools were not uni- 
form. Many o>f the greatest scholars and teachers of that 
period advocated the study of philology to the exclusion of 
history, geography, physics and mathematics. In his school 
laws Trotzendorf prescribed : " The students are never to use 
their mother tongue with teachers, fellow students or learned 
people." It was boasted of Trotzendorf s teaching: 2 "He 
has so thoroughly instilled the Latin language in all, that it is 
considered a disgrace to speak in the German language; to hear 
the servants speak Latin, one would believe that Goldberg 
lay in Latium." The school regulations 3 for Stuttgart in 

1 Unger, Die Methodik der praktischen Arithmetik, p. 1. Leipzig, 1888. 

2 Ibid., p. 5. 3 Ibid. 



THE ESSENTIAL FEATURES jyj 

1 501 commanded the school master to punish with scanty 
food the students who spoke German to> one another. Even 
Melancthon, who did much through lectures to encourage the 
study of mathematics, laid special stress upon language. 
" The school master, so far as possible, is to> speak nothing but 
Latin with the boys in order that they may become accus- 
tomed to it." The early Jesuit schools in France were like- 
wise closely confined to the Latin language. In England 
Roger Ascham, though at one time a lecturer in mathematics 
at Cambridge, expressed his views thus : * " Some wits, mod- 
erate enough by nature, be many times marred by overmuch 
study and use of some sciences, namely, music, arithmetic 
and geometry. These sciences, as they sharpen men's wits 
overmuch, so they change men's manners oversore, if they 
be not modestly mingled and wisely applied to some good use 
in life." But in all of these countries there were men who 
took a broader view of education, Martin Luther encour- 
aged the extension of the course of study. He said : 2 " If 
I had children, and if it were possible, they would not only 
learn language and history, but also music, singing and 
mathematics." Michael Neander, head of the Latin School 
of Ilfield, contrary to the practice of Sturm and Trotzendorf, 
taught geography, history and the natural sciences. 3 Pere 
Lamy, a leader of the Oratorians, exerted a similar influence 
in France. He said : 4 " I know of nothing of greater use 
than algebra and arithmetic." And Rabelais, 5 the great 
Benedictine, advocated the learning of mathematics " through 
recreation and amusement." 

In all of the Latin Schools the study of Latin consumed the 

most time. Mathematics seldom: extended over more than 

half of the course, and in some schools it was not taught 

| until the fifth or sixth year, and then only an hour weekly. 

1 The Scholemaster. 

2 Unger, Die Methodik, p. 5. 

3 Seeley, History of Education, p. 179. 

4 Compayre, History of Pedagogy, p. 151. 

5 Ibid., p. 98. 



1 72 SIXTEENTH CENTURY ARITHMETIC 

The plan of individual instruction used in these schools tended 
to discourage the teaching of arithmetic in the first years of the 
course. For each pupils had to prepare his lessons from a 
text-book in Latin, hence he had to master the language be- 
fore he could read his book. Classes which read Terence 
and Cicero and studied syntax and prosody, learned only the 
four operations with whole numbers. The full course sel- 
dom advanced beyond fractions and the Rule of Three. The 
following extract from a curriculum of a six-class Latin 
School of the sixteenth century gives some idea of the studies 
pursued : * 

" The practice of the sixth class ; figures and numbers ; of 
the fifth, common reckoning; of the fourth, not given; of the 
third, music, arithmetic and astronomy; of the two highest 
classes, second and first, not given." 

The following is from the curriculum of a five-class Latin 
School of the same period : 

" On Friday from 12 to i the arithmetic shall be read. The 
preceptors shall use no other arithmetic than that of Piscator 
(John Fischer), and in the fourth class (the next to the high- 
est in which Terence and Cicero are read) the species alone, 
in the fifth (highest class) the whole arithmetic is read.'" 

The courses of the early English Grammar Schools were 
very similar to those of the Latin Schools. About three- 
fourths of the time was given to the study of the classics. 
Few head masters paid any attention to mathematics. The 
Writing Master usually taught some arithmetic in the forms 
below the fifth and sixth. 2 

Since Latin was prescribed as the language of instruction 
in the Latin Schools, it was necessary to provide reckoning 
books in that language. Among the most important writers 
of these text books were: Michel Stifel, Hieronymus Car- 
danus, Gemma Frisius, Peter Ramus, Christopher Clavius, 
Simon Stevin, Piscator (John Fischer). 

1 Unger, Die Methodik, pp. 24-25. 

2 W. H. D. Rouse, History of Rugby School, p. 130. 



THE ESSENTIAL FEATURES iy^ 

Michel Stifel (1487-1567), an Augustinian monk, escaped 
from a monastery and went to Wittenberg in 1523, where he 
became a friend and follower of Luther. The last years of 
his life were spent at Jena as a private teacher of mathe- 
matics. Not only was Stifel interested in serious theological 
and mathematical subjects, but also in visionary fancies 
about the secrets of number. These fancies led him to write 
" Ein Rechen Biichlein von End. Christ. Apocalysis in 
Apocalysim, Wittenberg, 1532." He reckoned the last day 
of the world to be October 19, 1533, and imparted this know- 
ledge to his parishioners at Lochau, who consumed their for- 
tunes and goods. Stifel's life was endangered by this false 
prophecy and was saved only by the personal intervention 
of Luther. He then gave himself up to serious study and in 
1544 wrote "Arithmetica integra," a work which gave him 
a place among the leading mathematicians of the sixteenth 
century. 

Hieronymus Cardanus (1501-1576) taught mathematics in 
many Italian towns. He wrote " Practica arithmeticae gen- 
eralis et mensurandi singularis 1537," and "Ars magna arith- 
meticae. Artis magnae sive de regulis algebraicis liber unus, 

1 545-" 
Gemma Frisius (1 508-1 558) was Professor of Medicine at 

the University of Lourain and the author of "Arithmeticae 
Practicae Methodus Facilis, 1540." This book, small in com- 
pass, but rich in contents, was as popular in the Latin Schools 
as Adam Riese's was in the Reckoning Schools. It is com- 
posed of rules with an example to illustrate each. 

Peter Ramus (1515-1572), an anti-scholastic, was a pro- 
fessor at the University of Paris; but on account of his af- 
filiation with the Huguenots, he was obliged to leave France 
in 1560. He returned to Paris (1571) and was slain in the 
St. Bartholomew massacre (1572). Ramus was more im- 
portant as a philosopher than as a mathematician. He was 
the author of " Scholarum mathematicarum libri XXXI, 
Basel, 1569 " and "Arithmeticae libri duo" (1567.) 

Christopher Clavius (1537-1612), a Jesuit, became a teacher 



I 7 4 SIXTEENTH CENTURY ARITHMETIC 

of mathematics in the Jesuit College at Rome. He assisted 
in a revision of the calendar under Gregory XIII (1582), 
and wrote Opera Mathematica, 161 1," 5 vols., and " Chris- 
tophori Clavii Bambergensis e soeietate Jesu Epitome 
Arithmeticae Practicae, Rome, 1583." This was to be only 
the predecessor of a complete arithmetic, which, unfortun- 
ately, did not appear. In the subject matter of his works 
Clavius did not approach Tartaglia, but in methodical treat- 
ment was his peer. 

Simon Stevin (1548-1620) was a book-keeper at Antwerp, 
a revenue officer at Bruges, and a teacher and favorite of 
Prince Maurice of Nassau, by whom he was appointed in- 
spector of the dikes in Holland. He discovered decimal frac- 
tions and published the oldest discount table. His works 
were edited by Girard : " Les oeuvres mathematiques de 
Simon Stevin de Bruges. Le tout revu, corrige et augmente 
par Albert Girard, Ley den, 1634." The arithmetical part, 
written in the form of Latin compendia, is characterized by 
complete definitions and the development of processes from 
concrete problems. 

Piscator (John Fischer) was the -author of Arithmeticae 
Compendium (1545). The school regulation of Kursachen 
(1580) prescribed this arithmetic of Piscator to be used in the 
school. He also wrote reckoning books in the German 
language. 

Thus, the writers of Latin School arithmetics, with few ex- 
ceptions, were instructors in private schools, in Latin Schools, 
or in Universities. Their activities, however, were not con- 
fined to mathematics; for example, Ramus achieved fame as a 
philosopher; Frisius as a physician, and Stifel as a theologian. 
Hence, although the arithmetic of the Latin Schools was 
meagre, the text-books were written by the foremost schol- 
ars in medicine, science, engineering and philosophy. 

Since the essential features of the arithmetic produced by 
the above writers have been noted (pp. 23-170) it is necessary 
only to summarize them here. These works are characterized 
by a prominence of pure arithmetic, somewhat precise 



THE ESSENTIAL FEATURES ^ 

definitions (pp. 29-35), rules and explanations of processes 
with examples. Although one can recognize few principles 
of method consciously applied in mathematical instruction at 
that time, the arithmetics of certain Latin School writers con- 
tain material classified and organized with a view to facilitat- 
ing the educational process ; for example, the graded treat- 
ment of the cases in the multiplication of integers (p. 61), in 
the validity and sufficiency of the proofs of sevens, nines and 
elevens (pp. 44, 66), and the recognition of the relation of 
inverse processes (p. 109). The subject matter was usu- 
ally organized on the following plan : The whole realm of in- 
tegers was presented at the outset (pp. 36-77) and repre- 
sented under each operation. The four processes were re- 
peated with fractions and often with denominate numbers. 
There were no stages based on the size of numbers, no con- 
centric extension of operations, and no limits corresponding 
to the different ages of pupils. The following examples il- 
lustrate the rapid plunge into large numbers in the addition 
of integers : 1 



4 
3 


309 
204 


59 
34 


389 
204 


7389 
1264 


7389 20 
6264 10 


7 


13 


93 


93 


8653 


13653 30 


402 
301 


4052 
3601 




4321 
10 


730x55894 
60203643 


69001303 
69000000 
69008000 
















79017100 












79000003 
79006000 
89004000 
89026200 
89008000 
89005000 












99002400 












99017002 




1008095008 


thiber 


t Tonstall, 


De Arte Supputandi (1522), 


fol. C 3 recto, et seq 



176 SIXTEENTH CENTURY ARITHMETIC 

The applications of the processes given in the Latin School 
books were not numerous. The tendency was towards puz- 
zles, and factitious and traditional problems. For example, 
Ramus x gave the courier, the tower, statue of Pallas, the 
herds of Alcides, the mill-wheel, the fountain, the architect, and 
the widow problems (pp. 138-9). In all of the Latin School 
arithmetics there was a dearth of business problems. The 
rule of three, partnership, alligation, rule of false position and 
a few others were generally included, but the practical prob- 
lems characteristic of the commercial world of that time were 
generally ignored. The arithmetic of Gemma Frisius, which 
ran through some fifty-five editions, and which is probably the 
best example of Latin School text-books, contains for its ap- 
plications artificial problems of this kind : 2 

"A man having a certain number of aurei bought for each 
aureus as many pounds of pepper as equaled half of the whole 
number of aurei. Then upon selling the pepper he received 
for each 25 pounds as many aurei as he had at the beginning. 
Finally, he had 20 times as many aurei as he had at first. 
The number of aurei and the quantity of pepper are required." 

" Three men together have a certain amount of silver, but 
each one is ignorant o'f the amount he has. The first and 
second together have 50 aurei, the second and third, 70 aurei, 
the third and first, 60 aurei. It is required to know how 
much each one has." 

The reasons for teaching arithmetic in the Latin Schools 
must be deduced from indirect evidence, for discussions of 
courses of study or of educational values were so rare in 
that period that history gives little direct evidence on the sub- 
ject. One must look to the scholars and leading educators of 
that age, as of every age, and not to the directors and teach- 
ers of schools, to learn of the ideals and purposes of education. 
The scholars and educators of the sixteenth century were 

1 Peter Ramus, Arithmeticis Libri Duo (1586 ed.), fols. M 4 recto, M 5 
recto. 

2 Gemma Frisius, Arithmeticae Practicae Methodus Facilis (1540). 



THE ESSENTIAL FEATURES X yy 

enthusiasts for classical studies and the schools reflected their 
tastes. There was much of grammar, rhetoric, and literature 
in the Latin language, hence there was much of grammar, rhe- 
toric and literature in the schools. There was little of physi- 
cal science and mathematics available in Latin, hence there 
was little of these subjects in the schools. Greek mathema- 
tics, however, including the works of Euclid, Ptolemy, Archi- 
medes and Diophantus were of great interest to scholars, and, 
being early made available by translation, gradually found 
place among school studies. 

That the arithmetic of the classical languages did leave its 
stamp upon the works of the sixteenth century writers is evi- 
dent from the Greek classifications of numbers found in the 
extended treatments of proportion of the Latin School Arith- 
metics. Gemma Frisius treats at length harmonic and arith- 
metic proportion, proportion of equal and unequal numbers, 
multiplex, superparticular, superpartiens, multiplex superpar- 
ticulare and multiplex superpartiens proportion, proportion of 
fractions, mean proportional, and addition and subtraction of 
proportion. The extent to which the classics influenced Maur- 
olycus, a contemporary, is shown by this partial bibliographi- 
cal list : x 

Euclidis elementa. 

Theodosij Sphaerica elementa. 

Menelai Sphaerica. 

Apollonij Conica elementa. 

Sereni Cylindrica. 

Archimedis opera. 

Jordanj Arithmetica. 

Theonis Data geometrica. 

Rogerii Bacconis. & Io. Petsan Perspectivae breuiatae cum adnotation- 

ibus errorum. 
Ptolemei Specula. 
Autolyci de sphera. 
Theodosii de habitationibus. 
Euclidis Phaenomena brevissime demonstrata. 
Aristotelis problemata mechanica, cum additionibus complurimis, & iis, 

quae ad pyxidem nauticam, & quae ad Iridem spectant. 

1 Franciscus Maurolycus, Arithmeticorum Libri Duo (i575)> fol. Hh 
recto. 



178 SIXTEENTH CENTURY ARITHMETIC 

Therefore, one reason for the teaching of arithmetic in the 
Latin Schools was that it formed at least a 'small part of the 
classical inheritance. Another reason why arithmetic was 
taught in the Latin School was for its culture value. It was 
believed by the writers of arithmetical text-books to be an es- 
sential part of knowledge and of an education. The preface 
to nearly every arithmetic contains a eulogy on the importance 
of the science and its functions. The authors referred to the 
opinions of Greek philosophers, as Plato, who held that arith- 
metic alone distinguished men from the unreasoning beasts. 
They quoted from the Church Fathers and Holy Writ. They 
declared arithmetic to be valuable not only as an art, but as an 
essential to philosophical attainment and to an understanding 
of the mysteries of the Scriptures. That arithmetic was not 
taught in the Latin Schools in order to make proficient reck- 
oners is shown by the lack of practice problems in their text- 
books ; and, likewise, the lack of vital commercial problems of 
that day show that it was not taught in order to prepare for 
a business life. It is evident from such considerations that 
these scholars and writers neglected the applications of the sub- 
ject and believed theoretic arithmetic to be an essential in pro- 
ducing mental efficiency. 

Arithmetic in the Reckoning Schools 

The Reckoning School was born o>f necessity and owed its 
origin to the commercial development of the thirteenth cen- 
tury. The schools conducted by the clergy taught chiefly 
reading and writing. Instruction in book-keeping and reck- 
oning, the necessary equipment for a business career, was not 
found in the schools. Hence, merchants were obliged to in- 
struct their sons and to take others as apprentices who wished 
to enter commercial life. Under the direction of the Han- 
seatic League commerce grew so rapidly that merchants had 
to employ assistants to give instruction in reckoning. These 
teachers were the forerunners of the Reckoning Masters. 
As the merchant leagues increased in importance, municipal 
governments conferred upon them privileges relating to the 



THE ESSENTIAL FEATURES jjg 

management of trade; and this connection between govern- 
ment and private enterprise accounts for the official position 
and title, " Privileged Town Reckoning Master/' to which 
the teacher of commercial arithmetic arose. As a public offi- 
cer he usually acted as town clerk, inspector of weights and 
measures, shipping master, notary, and sometimes as surveyor. 
In his capacity as teacher he often monopolized the secular in- 
struction of his town. Thus, were founded the Reckoning 
Schools, which sprang up in the towns along the trade routes 
of the Hanseatic League, and, with certain modifications, throve 
in Italy, Germany, France, the Netherlands, and possibly in 
England. The Reckoning Master became such an important 
factor in education toward the end of the sixteenth century 
that he was even called upon to supplement the work of the 
Latin Schools. A record of appointment of a Reckoning Mas- 
ter at Rostock in 1627 contains the following: 1 "We, the 
Burgermeisters and members of the council at Rostock, an- 
nounce hereby that we have appointed the honorable and well 
educated Jeremias Bernsterz for our common Town Writing 
and Reckoning Master, until one party or the other shall give 
notice a half-year in advance. In this document we also com- 
mand him to attend the Latin School every week an hour each 
on Mondays, Tuesdays, Thursdays and Fridays and there to 
teach the youths without discrimination, and for a moderate 
monthly or weekly salary to teach others outside of the school, 
whether they be boys or girls who 1 desire instruction, Latin and 
German writing, reckoning, book-keeping and other useful 
arts and good manners ; he is also 1 to* do with greatest industry 
and according to the best of his knowledge and ability all 
other things which are properly the duties of an industrious 
and honest Writing and Reckoning Master. That his honest 
service may be properly rewarded, we shall pay him 400 
marks a year from the common treasury at four quarterly 
payments; we also promise him exemption from taxes, ex- 
cises, hundred penny tax, soldier money and all other con- 

1 Unger, Die Methodik der praktischen Arithmetik, pp. 26-27. 



180 SIXTEENTH CENTURY ARITHMETIC 

tributions, whatever they may be; we grant him the freedom 
of the city and a free dwelling-. We promise all in honor and 
good faith." 

As the number of Reckoning Masters increased, especially 
in the larger commercial centers, guilds, regulated by a con- 
stitution and by-laws, were formed. Boys who wished to be- 
come Reckoning Masters served an apprenticeship of six 
years, after which, by successfully passing a prescribed ex- 
amination and signifying their allegiance to the constitution 
of the guild, they were admitted to membership. There is 
preserved in Nuremburg school history the record of a diploma 
certifying that a candidate had passed the required examina- 
tion, July 12, 1620. 1 Among the signatures is that of Johann 
Heer, Reckoning Master and author of an excellent reckon- 
ing book (p. 12 Bibl.). Although these guilds were of great 
service in the sixteenth century in furnishing the common 
people a practical education, they later became a serious hin- 
drance to public instruction, because they were not obliged to 
follow the advances in educational methods. 2 

The function of the Reckoning Schools was to teach reckon- 
ing, business forms and the solution of commercial problems. 
But, since individual instruction from a text-book was the pre- 
vailing method of teaching, the pupil was obliged to read and 
write in his native tongue before he could study arithmetic. 

The courses of study, as verified by the above record of 
appointment consisted of the vernacular, writing, business 
forms, arithmetic, and sometimes Latin. Arithmetic was the 
chief subject, and language a secondary one, in contrast with 
the Latin Schools, where language held the major place and 
arithmetic a minor one. 

The authors of arithmetical text-books for Reckoning 
Schools wrote in German, Dutch or Italian to correspond to 
the language of their readers; an occasional one, as Piscator 
(John Fischer) also wrote books for the Latin Schools. 
Similar books for the use of merchants appeared in other coun- 
1 Unger, Die Methodik der praktischen Arithmetik, pp. 31-32, 33- 

2 Ibid. 



THE ESSENTIAL FEATURES ^i 

tries, as in France and England. The most important writers 
of reckoning books were : Ulrich Wagner, Johann Widman, 
Jacob Kobel, Adam Riese, Christopher Rudolff, Peter Apianus 
(Bienewitz), and Simon Jacob. 

Ulrich Wagner, a Nuremberg Reckoning Master, was the 
author of the first arithmetic published in Germany x (1482). 

Johann Widman entered the University of Leipzig with 
a certificate of poverty and took the degree of Bachelor of 
Arts in 1482 and that of Master of Arts in 1485 without cost. 
He became a teacher, lectured at Leipzig, and probably held 
a professorship. His chief work was his reckoning book : 
" Behend und hupsch Rechnung ufY alien Kauffmanschafften," 
Leipzig, 1489. Five other editions are known, Pforzheim, 
1500, 1508; Hagenau, 1519, 1521; Augsburg, 1526. 

Jacob Kobel (1470-1533) studied jurisprudence, mathe- 
matics and astronomy at Heidelberg. He made a special 
study of mathematics at Krakau (1490), where he was a 
fellow student of Copernicus. After his return to southern 
Germany, Kobel became town clerk at Oppenheim ( 151 1 ) 
where he passed the remainder of his life. He is best known 
as a poet, designer, wood carver, printer, publisher and mathe- 
matician. His fame as a reckoning book writer rests upon 
these works : " Eynn newe geordent Rechebiichlein vf den 
Linien mit Rechepfenigen (15 14); Das new Rechepuchlein 
Wie mann vff den Linien und Spacien/ mit Rechepfenning/ 
kauffmanschaft (1518)"; " Mit der kryde od' schreib- 
federn/ durch die Zieferzal (1520) ;" " Ein new geordet Visir- 
buch (15 1 5)." In 1 53 1 Kobel combined the last three books 
into one which was designed for self-instruction. 

Adam Riese, also Ries, Rys, Ryse (1492-1559), Germany's 
most famous Reckoning Master, taught at Erfurt (1522) and 
at Annaberg (1525). His reckoning books are: I. "Rech- 
nung auff der linihen gemacht durch Adam Riesen vonn Staf- 
felsteyn/ in massen man es pflegt tzu lern in alien rechen- 

1 A book published by the same printer, Petzensteiner (1483), probably 
was also a work of Wagner. This is known as the Bamberg Reckoning 
Book. 



1 82 SIXTEENTH CENTURY ARITHMETIC 

schulen gruntlich begriffen anno 1518/' Colophon: " Getruckt 
tzu Erffordt durch Matches Maler M. CCCCCXXV Jar." ; II. 
Reehenung auff der linihen vnd federn in zal/ mafs, vnd 
gewicht auff allerley handierung/ gemacht vnd zusamen 
gelesen durch Adam Riesen von Staffelstein Rechenmeister 
zu Erffurdt im 1522 Jar. Itzt vff sant Annabergk durch in 
fleyssig vibersehen/ vnd alle gebrechen eygentlich gerecht- 
fertigt/ vnd zum letzten eine hubsche vnderrichtung ange- 
hengt." III. " Rechnung nach der lenge/ auff den Linihen 
vnd Feder. Darzu forteil vnd behendigkeit durch die Propor- 
tiones Practica genant/ mit griintlichem vnterricht des visi- 
ren's. Durch Adam Riesen im 1550 Jar." Colophon: 
" Gedruckt zu Leipzig durch Jacobum Berwalt." IV. Ein 
Gerechent Biichlein/ auff den Schoffel/ Eimer vnd Pfundt 
gewicht/ zu ehren einem Erbarn/ Weisen Rathe auff Sanct 
Annaberg durch Adam Riesen, 1533. Zu Leiptzick hatt 
gedruckt diss gerechent Biichlein Melchior Lotter. Volendet 
vnd aufgangen am abendt des Newen Jars. 1536." The pic- 
ture of the author appears in his third book with the legend, 
"Adam Riese in the year 1550 at the age of 58," which fur- 
nishes the date of his birth. Riese's reckoning books were 
for a century the most popular books of the people and were 
reprinted many times in several combinations. 

Christopher Rudolff was educated at the Vienna Hoch- 
s'chule. His writings were a Coss (1525), Kiinstliche Rech- 
nung mit der Ziffer vnd mit den Zalpfennigen sampt der 
Wellischen Practica/ vnd allerley vortheyl auff die Regel de 
Tri/ alien Liebhabern der Rechnung vnd sonderlich dersel- 
bigen kunst anfahenden Schulern zu nutz/ Wien 1526," and 
"Exempel Buchlin" (1530). The second, a reckoning book, 
was his most popular work. Three later editions were pub- 
lished (1546, 1574, 1588). Rudolff was one of the few 
writers of that time who did not copy Riese. 

Peter Apianus (Bienewitz) (1495-1552), a scholar of great 
breadth, was educated at Leipzig and became the professor 
of Astronomy at the University of Ingolstadt and teacher of 
Kaiser Karl V. His interest included astronomy, geography, 



THE ESSENTIAL FEATURES 



183 



philology and mathematics. Besides a work on the Coss and 
one on astronomy, for which the emperor presented him 
with 3,000 gulden, Apianus wrote " Eyn newe vnd wo'lge- 
griindte vnderweysung aller Kauffmannfsrechnung in dreyen 
Buchern mit schonen Regeln vnd fragstucken begriffen, Sun- 
derlich was fortl vnd behendigkeit in der Welschen Practica 
vnd Tolleten gebraucht wirdt. Desgleichen fiirmalfs weder in 
Teutzeher noch in Welscher sprach nie gedruck. Durch Pe- 
trum Apianum von Leyfsnick d'Astrononiie zu Ingolstadt 
Ordinariu verfertigt." Colophon : " Gedruckt vnd vo- 
lendt zu Inglostadt durch Georgium Apium von Leyfsnick 
im Jar. nach der geburt Christi 1527 am 9. tag Augusti." 
Thus he deprived Grammateus of the honor of being the only 
University teacher who had written a German reckoning book. 

Simon Jacob of Coburg (died, 1564), was the most import- 
ant reckoning master of the last half of the sixteenth century. 
He wrote a reckoning book as well as a work on geometry. 

The first German work on book-keeping was written; by 
Heinrich Schreiber, (Grammateus). 

Corresponding to the German writers there was a school 
of writers of practical and commercial arithmetics in other 
countries. Among the Italians were the author of the Tre- 
viso book (1478), the first printed arithmetic, Philip Calandri 
and Piero Borgi ; among the Dutch a century later, Willem 
Raets, Jaques Van der Schuere, Martin Wencelaus and 
Ludolf van Ceulen; among the French, Savonne and Tren- 
chant ; among the Spanish, Juan Perez de Moya, Ortega, and 
among the English, Robert Recorde and Humphrey Baker. 
Most of the authors of reckoning school arithmetics, like the 
writers of Latin School arithmetics, were scholars of high 
rank. But they were in closer touch with the needs of the 
schools for which they wrote, for many of them served a9 
Reckoning Masters, while the Latin writers more often were 
professors in the Universities. 

It is noticeable that the contents of the reckoning school 
arithmetics were deficient in pure arithmetic. Little attention 
was given to definitions; the operations were stated dogma- 



1 84 SIXTEENTH CENTURY ARITHMETIC 

tically; few attempts were made to grade any operation into 
steps; and numbers larger than those commonly used in 
business were avoided. Numerous examples for practice were 
given in connection with the processes or in separate chap- 
ters. RudolfFs book, for example, was divided into three 
parts, Book of Elements, Book of Rules, and Book of Ex- 
amples. Much importance was attached to rules, and several 
writers resorted to verse in order to render them more effec- 
tive (p. 133). The following subtraction rule by Reichel- 
stain (1532) is a popular illustration: 

So du magst von der obern nit 
Ein ziffer subtrahirn mit sitt, 
Von zehen sollt sie ziehen ab, 
Der nechst under addir eins knab. 

Thus, the theory of arithmetic was treated less scientifically 
in these books than in the Latin School books. 

The reckoning book, in harmony with its purpose, was rich 
in applied problems. Problems involving denominate num- 
bers, exchange and merchants rules occupied the greater part 
(pp. 77-85, 132, 158). Besides commercial problems, men- 
suration received much attention, because the Reckoning Mas- 
ter, who often performed the duties of surveyor in his town, 
included in his reckoning book the methods for finding heights 
and distances (pp. 163-7) ; and, since as inspector of im- 
ports he had to measure the bales and casks, he gave in his 
books the theory and practice of the gauge. 1 It was usual 
for the applications to follow directly the explanation of the 
processes, although in several works a problem! was pro- 
posed before the process required had been taught (p. 48), 
a plan revived in the eighteenth century by Sturm and Wolf. 

The reasons for teaching arithmetic in the reckoning schools 
scarcely need to be stated. The origin and function of these 
schools and the interests of their teachers leave no doubt as 
to why arithmetic was taught. Utility was their shibboleth. 
They sought to make good computers and to give a thorough 
training in industrial and commercial arithmetic. 

1 Jacob Kdbel, Ein new geordet Visirbiich (1515)- 



CHAPTER II 

Educational Significance of Sixteenth Century 
Arithmetic 

We have set forth in Chapter I the contents of sixteenth 
century arithmetic. We have seen that it was produced by 
the leading scholars and teachers of that period, and that the 
chief influences responsible for this product were : the demands 
of the commercial and industrial world, the educational ideals 
and practices, the restraint of traditional customs, and the de- 
mands of science. We have seen that the purposes or aims 
•of the writers of arithmetic were : to furnish useful informa- 
tion, to provide material for mind training, and to advance 
human knowledge for its own sake. And, furthermore, since 
the leading spirits among the authors of arithmetic in that 
period were also the leading spirits among the teachers of the 
subject, the teacher's aims corresponded to those of the author; 
namely, to furnish the mind with useful information, to exer- 
cise it in mathematical thinking, and to interest it in arith- 
metic for the sake of knowledge getting. 

We recognize in these influences and aims the same forces 
and purposes that control the present day production and 
teaching of arithmetic. Hence, a study of the questions which 
arose and were solved at that time in relation to the corres- 
ponding questions now pressing for an answer will show to 
what extent we may profit from this inheritance; in other 
words, it will show the educational significance of sixteenth 
century arithmetic from the point of view of the present time. 

Both the problems of that period and those of to-day con- 
cerning the teaching of arithmetic may be classified under 
the heads: subject-matter, method, and mode. 

185 



186 SIXTEENTH CENTURY ARITHMETIC 

Subject-Matter 
We have seen that the subject-matter of arithmetic con- 
sisted of rules for reckoning with counters (line reckoning, 
pp. 25-26), rules for reckoning with the Hindu numerals 
(figure reckoning, pp. 27-29), a body of properties and rela- 
tion of numbers (pp. 29-77), a system of denominate num- 
bers (pp. 77-85) , and a heritage of amenity or puzzle problems 

(P- 159)- 

The chief question relating to* subject-matter which con- 
fronted sixteenth century writers was the basis of its selec- 
tion. The few who wrote in an encyclopedic fashion, Paciuolo 
and Tartaglia for example, tried to furnish arithmetic of all 
kinds and suited to all purposes ; but most writers grasped a 
special need or set up an. ideal and formulated their works in 
harmony with it. 

We have seen in the cases of Riese and Kobel and other 
reckoning masters that the basis of the selection of subject- 
matter was the need of the trader. 1 By them, processes and 
number relations were reduced to those only which were of 
use in solving commercial problems. The simple operations 
with comparatively small integers and fractions were briefly 
explained but elaborately applied to business questions; and 
denominate numbers, because of their variety and complexity, 
were made a prominent feature. In the cases of Cardan, Uni- 
corn, Ramus, and other Latin School writers the basis of selec- 
tion of subject-matter was the need of the scholar. Conse- 
quently, these authors gave a more elaborate treatment of 
processes covering a larger number field, emphasized theoretic 
subjects, like proportion, progressions, and roots, extended 
arithmetic to the solution of equations, retained the puzzle 
problems, but neglected the business applications. In the 

1 This is seen from such dedications as: "Prepared for merchants" 
(Borgi), "For business purposes" (Riese), "For merchants, bookkeepers 
and beginners" (Van der Schuere), "To the right iworshipfull, the gov- 
ernoure assistentes, and the rest of 'the companye of Marchants adven- 
turers : Humfrey Baker Lodoner wisheth helth with continuall increase of 
comodity by their worthy travail." Baker. 



EDUCATIONAL SIGNIFICANCE jgy 

cases of still others the basis of selection seemed to be a pre- 
ference for the traditional. What had attracted Nicomachus 
and other Greeks, what had seemed to Boethius worth while, 
appealed to these authors and caused themi to perpetuate and 
extend the fanciful classifications of numbers handed down 
from antiquity (pp. 32-35). 

The same ideas control the selection of material in the pre- 
paration of arithmetics^ to-day. The needs of the trader in- 
spire the " commercial arithmetic/' the needs of the scholar 
inspire the " disciplinary school arithmetic," and the influence 
of tradition is responsible for the retention in both of these 
books of much formal arithmetic long since obsolete. 

A comparison of the conditions which made the " Rechen- 
buch " of the sixteenth century, with the corresponding- con- 
ditions of to-day, indicates that we are entering upon a new era 
of commercial arithmetic ; for, when a nation passes from pas- 
toral and agricultural pursuits to those of manufacture and 
trade, the transition seems to be reflected in the subject-mat- 
ter of its arithmetic. It was true of the European nations 
of the fifteenth and sixteenth centuries. The fall of Con- 
stantinople (1453) and the consequent westward movement of 
Christian civilization initiated the educational and industrial 
awakening of Italy. This movement was reflected in the first 
printed arithmetic (Treviso, 1478) and in the subsequent 
Italian works of Calandri (1491), Borgi (1484), Paciuolo 
(1494), and Tartaglia (1556). The Hanseatic League by 
opening trade routes developed the commercial possibilities 
of Germany. The establishment of reckoning schools along 
these routes provided the opportunities by which the great 
German Reckoning Masters, Ulrich Wagner, Adam Riese, 
and Christopher Rudolff rose to fame. The spirit of enter- 
prise invaded France in 1553, and Savonne ushered in com- 
mercial arithmetic. The Netherlands awoke to her maritime 
possibilities under the Inquisition through the oppression of 
the Spanish king, and practical arithmetic immediately found 
expression in the works of Van der Schuere and Raets. Eng- 
land, feeling the pulse of continental commerce, increased her 



188 SIXTEENTH CENTURY ARITHMETIC 

trade, and Recorde and Baker founded her practical arith- 
metic. The conditions in America to-day are somewhat 
similar to those of the European countries of the sixteenth 
century. The United States has built up a vast industrial sys- 
tem, on the basis of its natural resources. It has recently 
taken the position of a world power through commercial and 
political expansion, and the teaching of arithmetic is just be- 
ginning to respond to this movement. Teachers and educa- 
tors are demanding that the subject-matter of arithmetic shall 
meet the needs of the present. The school cannot prepare for 
life unless it reflects the interests of the people. It must re- 
cognize in its teaching the quantitative aspect of such far- 
reaching enterprises and industries as our railways, steam- 
ships, telegraphs, telephones, mines, quarries, ranches, plan- 
tations, and factories. The following is a broad classification 
of the type of problems that are beginning to appear in the 
arithmetics of the twentieth century : 

Problems of Resources : Products of soil and mine ; supply 
of timber and water power; capacity of development and 
consumption. 

Problems of Industries : Labor and capital, production of 
food products, clothing, tools, and luxuries. 

Problems of Transportation : Express rates, freight rates 
by railroad and steamships, capacities of carriers, and storage. 

Problems of Communication: Postal rates, telephone, tele- 
graph, and cable rates. 

Problems of Government: Expenses, revenues, enterprises, 
imports and exports, immigration, and employees. 

Problems relating to Education : Schools, libraries, print- 
ing, and cost of public instruction. 

Problems of Science and Art : Domestic economy and nature 
study. 

This, however, does not mean that twentieth century arith- 
metics are to be a revival of the Reckoning Books of the six- 
teenth century. The Reckoning Book owed its existence to 
the exclusion of commercial arithmetic from the schools of the 
Church, the Municipality, and the Teaching Order. No com- 



EDUCATIONAL SIGNIFICANCE T gg 

promise was possible, the educational ideal of the schorl men 
of that day was too limited to appreciate the needs of the 
merchant. Modern systems of education on the other hand 
welcome utilitarian arithmetic. The business men of to-day 
are not obliged to found special schools and seek special books 
in order to promote elementary business knowledge. The 
business college is a convenience and short cut to certain vo- 
cations rather than a substitute for public school instruction. 
Modern education tends toward uniformity and toward a cor- 
relation of interests, hence twentieth century arithmetic will 
not be wholly utilitarian like that of the Reckoning Books, 
but will be a composite of materials chosen on all three bases, 
the needs of the business man, the needs of the scholar, and 
the influence of tradition. 

Another modification of the subject matter of arithmetic in 
the sixteenth century due to the commercial ideal was the 
transition from line (abacus) reckoning to figure reckoning 
(algorism). This change in the subject matter O'f arithmetic 
was hastened by the conviction that the Hindu reckoning was 
a more efficient system of calculation than the abacus in solv- 
ing business problems. The abacus was not discarded because 
it was a machine, but because a better method was found to 
do its work. A similar tendency exists to-day through the 
introduction of the calculating machine. Although the mod- 
ern tendency is to change from head work to automatic ma- 
chinery, which seems to be exactly opposite to the change in 
the sixteenth century, it is, nevertheless, similar in being a 
change from the difficult to the more easy method of practical 
calculation. If the cost of a reliable machine for performing 
the four fundamental operations did not prevent its use from 
becoming universal, one might reasonably predict the dis- 
appearance of the formal processes from business arithmetic 
before the end of the present century. But this is certain, 
that the mechanical calculator by decreasing the need for 
expert accountants will lessen the amount of abstract drill 
matter in arithmetic and will limit the treatment of the oper- 
ations to the field of small numbers. 



I 9 SIXTEENTH CENTURY ARITHMETIC 

The needs of the trader affected the selection of subject 
matter in another way. No business arithmetic could be effi- 
cient without a thorough treatment of contemporary denom- 
inate numbers, and a glance at the practical arithmetics of the 
sixteenth century conveys the impression that an elaborate and 
exhaustive treatment of such numbers was essential. But a 
careful inquiry into the weights, measures, and moneys of that 
time shows that the variety in use was so 1 great and the 
awkwardness of the systems so extreme that even a con- 
servative exposition of them occupied a large share of the 
text-book (pp. 83, 147). This necessary prominence given to 
denominate numbers in the first printed books initiated a cus- 
tom which for centuries led writers to 1 compile tables of weights 
and measures long after they had ceased to be useful. The les- 
son which the sixteenth century has for modern writers on this 
point is that they should present only the contemporary sys- 
tems, taking full advantage of their simplicity and uniformity. 

We have shown what lessons concerning the selection of the 
subject matter of arithmetic may be drawn by comparing the 
influence of the sixteenth century commercial renaissance with 
that of the present industrial development. Similarly, we may 
obtain further suggestions by comparing the effects of the 
educational ideal then prevalent with the effects of our own. 
The scholar's need of arithmetic at that time was thought to de- 
pend upon three things, mental discipline in the narrow sense 
of logical thinking, culture in the broad sense of information 
and pleasure ini knowledge getting, and propaedeutics, or pre- 
paration for scientific research. 

The teaching of arithmetic at the opening of the twentieth 
century is in much the same condition that it was at the be- 
ginning of the sixteenth. The nineteenth century was an 
epoch of mental discipline, and just as the Latin Schools in- 
herited their ideal of formal studies, like astronomy and arith- 
metic from the Ouadriviumi of the Middle Ages, so the men- 
tal disciplinarians of the nineteenth century inherited their 
ideal from the Latin school of the Renaissance. 

It is true that the functions of school studies had not been 



EDUCATIONAL SIGNIFICANCE 191 

so thoroughly differentiated then as now, the time for begin- 
ning -the study of the classics had not been advanced to the 
period of secondary instruction, and therefore, arithmetic re- 
ceived a comparatively small part of the time given to school 
work. But the books inspired by the disciplinary idea had a 
distinctive content (pp. 174-176), operations extended over a 
large number field, lack of commercial problems, prominence 
of definitions and principles, and extreme classifications. 
These same features characterize the disciplinary arithmetics 
of to-day; and, in the light of history, should it not be so? 
It should, other things being equal, but they are not. We 
have a clearer and more correct idea o<f the disciplinary value 
of arithmetic now than schoolmen had then. We know that 
this value does not depend so much upon complex processes, 
definitions, principles, and minute classifications as upon the 
mathematical reasoning found in the applications of the sub- 
ject. " Mathematics is the science which draws necessary 
conclusions." 1 "According to this view, there is a mathema- 
tical element involved in every inquiry in which exact reason- 
ing is used, and one is not justified in calling reasoning 
mathematical unless it is exact." 2 An appreciation of the 
mathematical element in elementary arithmetic is chiefly gained 
by using the syllogism in solving concrete problems. In fact, 
the simple applications of arithmetic are the most available and 
sufficiently simple material for children to use as exercises in 
logic. For example, the second grade pupil who solves the 
problem, If one orange costs 2 cents what do five oranges cost ?, 
traverses the steps of the syllogism. For the minor premise 
is, one orange costs 2 cents; the major premise is, 5 oranges 
cost 5 times as much as 1 orange; and the conclusion is 5 
oranges cost 5 times 2 cents, or 10 cents. The complex prob- 
lems o>f more advanced arithmetic are generally composed of 

1 Benjamin Peirce, Linear associative algebra (1870); also American 
Journal of Mathematics, vol. 4. 

2 Maxime Bocher, Bulletin of the American Mathematical Society, Vol. 
XI, Number 3 (1904), p. 117. 



192 SIXTEENTH CENTURY ARITHMETIC 

a series of steps, each of which is solved by the application of 
the syllogism, as in the simplest case. 

For mind training the concrete side of arithmetic is more 
efficient than the abstract, because there is an unlimited amount 
and variety of the former, simple in point of logic, while there 
is little of the latter, whose logic is not beyond the elementary 
pupil's powers. Besides having variety the concrete side pos- 
sesses interest gained through the correlation of arithmetic 
with other school subjects and through the interpretation of all 
environment in its quantitative aspect, whereas the abstract is 
usually uninteresting to children. 

It has still another advantage over the abstract in that it 
deals with the same material which is used in other school 
subjects where the reasoning does not relate to quantity, al- 
though it is till mathematical. We are accustomed to say 
that arithmetic is the only subject from which the elementary 
school pupil can obtain practice in mathematical reasoning. 
But, " Mathematics does not necessarily concern itself with 
quantitative relations, any subject becomes capable of mathe- 
matical treatment as soon as it has secured data from which 
important consequences can be drawn by exact reasoning." 1 
Thus, all of the following, though drawn from different sub- 
jects, are examples of mathematical reasoning: 

i. Geography, a. There is cheap water power wherever 
there are rapidly flowing rivers. 

b. There are rapidly flowing rivers in New England. 

c. Therefore, there is cheap water power in New England. 

2. Applied Arithmetic, a. The ratio of the circumference 
to the diameter of any circle is w. 

b. The diameter of a circle is 3 ft. 

c. Therefore, the circumference of the circle is w -3 ft. 

3. Pure Arithmetic, a. A fraction whose numerator is 
greater than its denominator is an improper fraction. 

b. In I the numerator is greater than the denominator. 

c. Therefore, | is an improper fraction. 

1 M. Bocher, Bulletin American Mathematical Society, Vol. XI, No. 3. 
p. 118. 



EDUCATIONAL SIGNIFICANCE ! 93 

4. History, a. Oppression O'f a people means revolution, 
and revolution means democratic institutions. 

b. There is oppression in country A. 

c. Therefore, there will be democratic institutions in coun- 
try A. 

5. Pure Mathematics, a. An A-object is a necessary and 
sufficient condition for a B-object, and a B-object is a neces- 
sary and sufficient condition for a C-object. 

b. a is an A-object. 

c. Therefore, there is a C-object. 

It is sometimes claimed that arithmetic affords better dis- 
cipline in reasoning than does history, because the data in his- 
tory are subject to exceptions and lacking in generality. 
Thus, examples 2 and 3 above would be regarded as having 
greater educational value than example 4, because oppression 
has not always led to revolution, nor revolution to democratic 
institutions. But mathematical reasoning is not concerned 
with the validity of its data; therefore, as an exercise in mathe- 
matical logic, example 4 is in no way inferior to the others. 
If the soundness of the data be considered, the studv of his- 
tory would seem, to be more disciplinary than that of arith- 
metic. For, in arithmetic no forethought can be exercised in 
selecting true data, since the data are determined a priori. 
From this point of view modern grammar school teaching, 
which encourages exact reasoning from cause to> effect in re- 
gard to data drawn from history, from geography, and from 
industrial and domestic science is really securing the essence of 
mathematical discipline through varied symbols of expression. 

Thus it appears in selecting the subject-matter of arithmetic 
for disciplinary purposes that the concrete side should have the 
preference, because it furnishes material suited to easy logical 
thinking; material which is varied, abundant, interesting, and 
related to other school subjects. 

The conviction that arithmetic supplies a certain kind of in- 
formation needed in non-technical daily life has never in a 
large way influenced writers in the selection of subject matter. 
The probable reason for this is that the average citizen, not 



194 



SIXTEENTH CENTURY ARITHMETIC 



engaged in a trade or business especially requiring a know- 
ledge of arithmetic, uses very little of the subject. 1 Ancient 
peoples like the Chinese, Hindus, and Hebrews, who con- 
ceived of no special reason for teaching arithmetic used the 
merest elements ; practically all of the arithmetics of the six- 
teenth century agreed in the material essential to the needs of 
the common people, and similarly, all arithmetics of to-day 
agree in the subject matter pertaining to this purpose. But, 
the broader idea of information giving, that of presenting the 
subject for its own sake, and the broader idea of knowledge 
getting, that of learning for the pleasure of knowing, have 
greatly influenced the selection of the subject matter of arith- 
metic. The Latin School books (pp. 174-176) of the six- 
teenth century show the effect of this influence. As a rule 
they contain a more complete list of topics than do the Reck- 
oning School books (pp. 183-184). They often contain, in 
addition to a rounded treatment of the theory of the subject, 
various practical rules, much material of historical interest 
(pp. 129-131), and recognize the amenities of the subject 
(p- I 59)- It i' s the idea contained in the last two 1 particulars 
which throws light on the selection of the materials of present 
day arithmetic. 

Although it is customary to place the educational doctrine 
of interest at a very recent date, and correctly so in the sense 
of a formally stated theory, the essence of this idea has long 
been appreciated. It was this central thought of education 
that led the Latin School writers of the sixteenth century to 
humanize arithmetic by applying it to historical data, and by 
emphasizing the amenities of the subject. There is reason to be- 
lieve that this idea was strong enough to have thoroughly vital- 
ized the Latin School arithmetic had there been such subjects 
as Commercial Geography, Domestic Art, and Nature Study 
in the schools, available for correlation. Furthermore, that 
these authors, although necessarily unaware of the results of 
modern researches into child nature, grasped the importance 

1 David Eugene Smith, The Teaching of Elementary Mathematics, page 
21, New York, 1900. 



EDUCATIONAL SIGNIFICANCE IO c; 

of the play or game element in education, appealed to curiosity 
through magic squares and other remarkable properties of 
number, 1 and recognized the sociological phase through prob- 
lems touching social usages and conditions. One would seem 
justified in concluding that, since men while groping after edu- 
cational doctrine selected a part of the subject-matter of arith- 
metic on the basis of interest, writers to-day under the present 
flood of educational ligfat should consult the natural instincts 
of the individual in selecting such material. 

There is evidence of a basis for number in the sense of 
rhythm and the tendency to repeated action early manifested in 
the child. The tendency to imitate the doings of others cor- 
relates the simple muscular activities into more complex move- 
ments and enables the child to participate in games and occu- 
pations which have a quantitative side. The occupations and 
industries of the neighborhood, facts about food, clothing, and 
shelter, besides furnishing an abundance of material, make a 
basis for instruction in larger interests. In passing beyond 
the simple facts of home life, as the pupil's experience grows, 
the whole range of production, transportation, communication, 
and consumption, both local and national, are available as in- 
teresting applications of arithmetic. After leaving the prim- 
ary school the pupil enlarges his social interests. His ten- 
dency to hero worship influences his intellectual tastes, and 
the time is right for problems about fire and police protection, 

1 E. g., Unicorn, De L'Arithmetica vniversale (1598), mentions this curi- 
osity as taken from Frate Luca (Paciuolo), the point of interest being the 
repetition of the same digit in the product. 

777 777 

2 143 



1554 
143 


2331 
3108 

777 


4662 

6216 

1554 


iiiiii 



222222 



196 SIXTEENTH CENTURY ARITHMETIC 

about the army, the navy, the life-saving service and similar 
things of local or national concern. Even the ethical phase 
of instruction is not unrelated to arithmetic, for acts of charity 
and self-sacrifice in the school, the neighborhood, and the 
state dignify the problems of economics. 

Thus we have seen that the early printed arithmetics in their 
culture phase were prophetic of the important role which the 
subject is to play in education. 

The selection of arithmetic according to its propaedeutic 
value relates to the scholar's need of a basis for higher mathe- 
matical study. The sixteenth century possessed some of the 
greatest mathematicians of all time, some who did the world 
an inestimable service by advancing the science of mathema- 
tics. It is notable 'that such men as Tartaglia, Cardan, and 
Rudolff placed at the foundation of their works a well rounded 
treatment of theoretic arithmetic. It is probably true that 
mathematics for the sake of its presentation and advancement 
as a science requires a theoretic basis in arithmetic which 
should be considered in the selection of material for the pur- 
pose of its study. 

Finally, there should be considered in the selection of the 
subject-matter of arithmetic the influence of tradition. The 
effect of this influence may not be always entirely separable 
from that of either the needs of the business man or from 
those of the scholar, for many business customs reach far 
back into history, and the scholar's interest pertains to all 
ages, but it has one characteristic, namely, the tendency to per- 
petuate things regardless of their usefulness or interest. Just 
as certain arithmetics of the sixteenth century contained ma- 
terial which was there simply because it had long been in 
arithmetics (pp. 32-34, 159), so some text-books of to-day con- 
tain matter whose presence has no other justification. If we 
adopt the theory that a sufficient amount of mental discipline 
can be obtained from utilitarian subject-matter, then the fol- 
lowing items may well be omitted from twentieth century 
arithmetic. 



EDUCATIONAL SIGNIFICANCE igy 

Concerning Integers : 

Greatest common divisor of large numbers and the 

Euclidean method. 

The need for this process disappeared in the seven- 
teenth century with the introduction of the notation of decimal 
fractions. Before that time it was necessary to find the great- 
est common divisor of the terms of a fraction in order to re- 
duce it to its lowest terms (pp. 95-96). 

Least common multiple of large numbers, or of those 

not readily factored. 

The need for this process is confined to reducing frac- 
tions to a common denominator, but there is no actual demand 
for the treatment of fractions whose denominators are large 
numbers not readily factored. 

Cube root, especially of large numbers not readily 

factored. 

There is scarcely any excuse for teaching the general 
process of extracting cube root in the grammar school, or, in 
fact, square root, although the latter is sometimes met in 
practical measurement. The elementary school pupil seldom 
understands the reasoning involved in the process and there- 
fore loses even the supposed mental training. 

Progressions. 

These subjects have no vital applications, and their 
theory belongs to algebraic analysis. 
Concerning Fractions : 

Complex fractions. 

Drill work in fractions above thousandths. 

All fractions not common in business practice, as 39ths, 

47ths, 6ists, and the like. In such matters the ques- 
tion is not so much one of omission as of emphasis. 

The presence of such fractions in arithmetic may even 

be desirable, but continuous drill work with them is 

deadening. 
Concerning Decimals : 

Decimals beyond thousandths should not receive 

emphasis. 



198 SIXTEENTH CENTURY ARITHMETIC 

All practice in circulating decimals. 
Reduction of decimals to common fractions should re- 
ceive little emphasis. 

Concerning Denominate Numbers : 
Troy weight. 
Apothecary's weight. 
Surveyor's measure. 
Duodecimals. 

Not only should all obsolete tables be eliminated, but 
also those which are of use to specialists only. 
Gill, perch, mill, and rod. The last is still spoken of 
in rural districts, but is seldom met in practical 
calculation. There is no defence for its great promi- 
nence in text-book exercises. 

Tables of English Money should receive less attention 
than the exchange value of the pound sterling, the 
mark, and the franc. 

Compound numbers of more than two or three de- 
nominations are not justifiable in operations. 

Concerning the applications of arithmetic: 

Profit and Loss as a separate subject. A few prob- 
lems to illustrate percentage of gain and loss are 
sufficient. 

True discount. Bank discount has taken its place en- 
tirely. 

Partial payments in the form of state rules and irre- 
gular indorsements. Promissory notes which provide 
for payment before maturity are drawn with the privi- 
lege of paying stated amounts on interest days. 
Annual interest, except in the form pf bond coupons. 
Equation of payments. 

Partnership with time. Partnership without time may 
serve as an introduction to the explanation of the mod- 
ern Stock Company. 
Alligation. 

Compound Proportion. Simple proportion is import- 
ant, but the old form and the traditional processes of 



EDUCATIONAL SIGNIFICANCE I99 

inversion, alternation, composition, and so on, are be- 
ing- replaced by the simple equation. 
Besides these whole topics there are many types of problems 
which have become obsolete. The chief ones are: 

Problems in commission that represent the agent as re- 
ceiving money from his principal after deducting his 
commission. 

Problems in- stocks that represent the purchase and sale 
of fractional shares of stock. 

Problems in exchange that represent all rates of ex- 
change at par and that do not conform to modern 
banking customs. 

Problems in compound interest, apart from some bank 
or life insurance reckoning in which tables are 
employed. 

Then there are the objectionable inverse problems 
found everywhere, as : Given the proceeds, rate, and 
time to find the face. Given the rate and amount to 
find the base. Given the rate, time, and interest to- find 
the principal. Such problems may occasionally be per- 
mitted for the purpose of jogging the intellect, but 
seldom on the ground of utility. 
Having discussed the bases of selection of subject-matter, 
the organization o>f materials may next be considered. Al- 
though any arrangement of subject-matter must have a bear- 
ing upon method, it may be made for logical rather than for 
educational reasons. In the making of sixteenth century arith- 
metics there were three plans of construction. The first and 
most common was the prevailing nineteenth century type. 
The whole realm of numbers was presented at the outset 
(pp. 36-77) and was represented under each operation. The 
four processes were repeated with fractions and often with de- 
nominate numbers. Another plan, best illustrated by Cardan, 1 
was based upon the idea of teaching all the operations with 
each kind of number. The following outline of the first few 
chapters will illustrate: 

1 Cardan Practica Arithxnetice (1539). 



200 SIXTEENTH CENTURY ARITHMETIC 

Chapter I On the Subjects of Arithmetic. 

The subjects defined are: integral numbers, 
fractions, surds, and denominate numbers. 

Chapter II On Operations. 

There are seven operations : numeration, ad- 
dition, subtraction, multiplication, progres- 
sion, division, and the extraction of roots. 

Chapter III Numeration of Integers. 

Chapter IV Numeration of Fractions. 

Chapter VI Numeration of Denominate Numbers. 

Chapter VII Addition of Integers, including denomin- 
ate numbers. 

Chapter VIII Addition of Fractions. 

Chapter IX Addition of Surds. 

Chapter X Addition of Powers. 

Chapter XI Subtraction of Integers and Denominate 
Numbers. 
The third plan was a modification of these two in which a sub- 
ject like denominate numbers occurred under two or three 
operations. 

Although these schemes of arrangement were not based on 
avowed educational principles, they are none the less sugges- 
tive at the present time. Educators are generally agreed that 
the extreme topical plan of nineteenth century arithmetics is 
defective, and are seriously questioning the so called spiral 
plan, hence it may be useful to know by reference to six- 
teenth century arithmetic that there is no practical arrange- 
ment except a combination of the first and second plans de- 
scribed above. The scheme of grading the subject according 
to the size of the numbers and complexity of processes, instead 
of by kinds of numbers or kinds of processes seems to be an 
essential for primary teaching, but its value depends upon the 
size of the groups and upon the systematic treatment within 
each group. Some writers have chosen a group of topics, 
reproducing them every ten pages and adding slowly to the 
content presented. Some have chosen a group to represent a 
term's or a year's work, reproducing and extending the con- 



EDUCATIONAL SIGNIFICANCE 2 OI 

tent in each succeeding period. Others have reduced the 
amount of repetition to a minimum by giving only two cycles, 
one for the primary school and one for the grammar school. 
The proper number of periods lies between these extremes and 
probably approaches the larger rather than the smaller cycle. 
The minimum cycle leads to confusion and scrappiness, while 
the larger cycle makes possible the systematic treatment of ad- 
dition, subtraction, multiplication, and division separately un- 
der each kind of number within each group. It not only 
makes possible the appearance of these processes, but it in- 
sures to each an extent of application sufficient to leave an 
impression oi its importance upon the pupil. 

Method 

Method as used in this article relates to an author's manner 
of treatment of subject-matter and not to the teacher's mode 
of presentation of it in the class room. It is by considering 
method in this aspect that one obtains most light from sixteenth 
century sources on the teaching of arithmetic, because the text- 
books constitute the chief source of information concerning 
the mathematical instruction of that time. Even from this 
narrower point of view one cannot expect to gain informa- 
tion on all present questions of method, since the text-books 
of that time were written before much of modern educational 
doctrine saw the light. But there is enough of suggestion to 
justify a comparison of the methods of development of subject- 
matter at that time with those prevalent to-day, and for this 
purpose we may consider three plans, the synthetic, the analy- 
tic, and the psychologic. 

The synthetic method is the form in which the mathema- 
tician casts the finished product of his reasoning. It rarely is 
the way by which he reaches the truth, but having reached it, 
the synthetic development of the steps in the process makes 
an elegant and direct exposition. 

The analytic method is the process of experimentation and 
discovery. It is the pulling to pieces and comparison of parts 
which show on what principles the system may be built up. 



202 SIXTEENTH CENTURY ARITHMETIC 

The psychologic method seeks to develop the subject along" 
the path of least resistance by taking into account all accepted 
educational principles. This method is not wholly separate 
from the other two any more than logic is separate from psy- 
chology. In general, it conforms now to one and now to the 
other; but its characteristic feature consists in this, that it fol- 
lows psychological principles, whereas the other methods fol- 
low the canons of logic. 

All of these types of development are represented among 
sixteenth century arithmetics. The synthetic type, however, 
was the common method, the others being exceptional. Al- 
though its use was not confined to one school of writers, it pre- 
vailed almost exclusively among those of the Latin School. It 
was the traditional method of the ancient mathematicians, and 
the great classical scholars of the sixteenth century, as 
Paciuolo, Cardan, Ramus, Tartaglia, and Unicorn, preferred 
its elegant form of expression. Of course, arithmetic does 
not furnish opportunities for demonstrations to the same ex- 
tent that geometry does; there is more of definition and me- 
chanical process; but the spirit of the synthetic method is 
basal in all dogmatic treatments and in all topical arithmetics 
in which abstract theory precedes the concrete matter. This 
is the type represented by the old style books in use to-day. 
It is a relic of the old education in which adult psychology 
stood in lieu of child psychology and information giving in 
place of self-realization. 

The sixteenth century arithmetics which may be called 
analytic Avere cast in the catechetical form, a style somewhat 
prevalent for half a century. Among the first of these books 
was one by Willichius (1540), but it was not a very success- 
ful arithmetic. Perhaps the best extant example of this 
method is Recorde's Ground of Artes. The following is an 
illustration taken from his clear method of teaching the signi- 
ficance of place value in notation : 

" M. (Master) Now then take heede, these certayne values 
every figure representeth, when it is alone written without 
other figures joyned to him. And also* when it is in the first 



EDUCATIONAL SIGNIFICANCE 203 

place though many others doo follow : as for example. This 
figure 9 is IX. standing now alone. 

" Sc. (Scholar) How, is he alone and standeth in the mydle 
of so many letters ? 

" M. The letters are none of his felowes. And if you 
were in Fraunce in the middle of M. Frenchmen, if there were 
none Englysh man with you, you wolde recken your selfe to 
be alone. 

" Sc. I perceaue that." fol. Bvii verso. 
The catechetical method is one of those extreme forms of pre- 
senting arithmetic which has been tried and found wanting. 
But the lesson for present purposes is not that arithmetics 
should not be analytic in form, but that in making them 
analytic, there should be no attempt to usurp the function of 
the teacher. The ideal arithmetic must be inductive and must 
suggest ways of presenting subjects in the class-room, but all 
arithmetics which may be designed to serve both as text-book 
and teacher will prove unsuccessful as were the catechetical 
ones of the sixteenth century. 

Among the sixteenth century writers who were less en- 
slaved to the classical tradition, that is, the authors of the 
practical arithmetics of that period, are found those whose 
works represent the third or psychological method of develop- 
ment. For, although these writers were practically bound by 
prevailing conditions to dogmatic instruction, and, although 
they lived before the time of genetic pschology, they were not 
without educational sense. 

A noted example is that of Riese (p. 181) who recognized 
the following principles: 

1. From the concrete to the abstract — by placing reckoning 
with counters before reckoning with figures. 

2. From the simple to the complex — by presenting the full 
form of processes before the abridged form. 

3. Repetition, practice makes perfect — by working over the 
same material in different forms. 

It is significant that the authors who possessed this psycho- 
logical insight belonged to the practical school rather than to 



204 SIXTEENTH CENTURY ARITHMETIC 

the disciplinary school, and that the most useful and popular 
books of that century were those conforming to their method. 
The man with the greatest educational grasp, Adam Riese, 
was the greatest reckoning master and wrote the greatest reck- 
oning book of his time. This is exactly in accord with what 
promises to be true in the twentieth century, namely, that the 
ideal text-book will be the utilitarian arithmetic constructed 
according to the psychological method. 

Besides the results of these general comparisons relating to 
the organization of subject-matter, sixteenth century arithmetic 
set various precedents and is suggestive in respect to many de- 
tails. The educational significance of these particulars will 
best be expressed by grouping them in four classes : those 
pertaining to* definitions, to symbolism, to processes, and to 
applications. 

We have already noted that formal definitions, although not 
abandoned by any class of writers, were especially emphasized 
by the Latin School men. It was their text-book that set the 
practice, characteristic of all disciplinary arithmetics, of defin- 
ing terms and processes at the outset. But the style of the 
definition is significant. Thus, addition was generally defined 
as the collection of several numbers into one sum (pp. 35-36) 
in place of the process of -finding the sum of two numbers. 
Subtraction was generally defined as taking a smaller number 
from a larger one, which is more suggestive than the defini- 
tion, the process of finding the difference between two numbers. 
Certain writers improved on this and even recognized sub- 
traction to be the inverse of addition. Multiplication was 
generally defined as repeating one number as an addend as 
many times as there are units in another, a better definition 
than the process of finding the product of two numbers, al- 
though it is not generally applicable to fractions without modi- 
fication. Division was generally defined as finding how many 
times one number was contained in another, the partitive phase 
often being included, a more definite form than the process of 
finding the quotient. 

It is true that sixteenth century definitions were not general, 



EDUCATIONAL SIGNIFICANCE 



205 



nor can such definitions be used in elementary work. A 
statement which tells nothing and one which states an im- 
portant truth unintelligibly are equally useless. It is this fact 
which has led modern teachers to the extreme of wholly ne- 
glecting definitions. But the power of expressing thought is 
too significant to be crippled. Descriptions of a process or of 
the characteristic property of a term leads to clearness and 
efficiency. Sixteenth century arithmetic points to the conclu- 
sion that all definitions used in teaching should be working de- 
finitions, that is, they should state how the process is per- 
formed. It is possible to formulate statements of this kind that 
are not beyond the experience of the elementary school pupil, 
and which are sufficiently general to* admit of extension as new 
fields of mathematics are entered. 

Any phase of the growth of mathematical notation is an in- 
teresting study, but the chief educational lesson to be derived 
is that notation always grows too slowly. Older and in- 
ferior forms possess remarkable longevity, and the newer and 
superior forms appear feeble and backward. We have noted 
the state of transition in the sixteenth century from the Ro- 
man to the Hindu system of characters (pp. 24-26), the intn> 
duction of the symbols of operation, -f-, — , (pp. 53-55) and 
the slow growth toward the decimal notation (pp. J2, 75). 
The moral which this points for twentieth century teachers 
is that they should not encourage history to repeat itself, but 
should assist in hastening new improvements. For example, 
the use of x for (?) in equations, singular abbreviations, as lb. 
for lbs. ; and the use of % for per thousand ; 2 /z for f ; $ for $ ; 
and a convenient abbreviation of denominations of the metric 
system. 

No mention of Roman notation can pass without reference 
to the question, Why are we teaching it to-day ? Less than a 
decade ago 1 , it appeared that the traditional clock-face and the 
numbering of introductory pages and chapters with Roman 
numerals were doomed. But the recent craving for antiques 
and its influence on certain crafts are making a knowledge of 
Roman notation to thousands still desirable. 



206 SIXTEENTH CENTURY ARITHMETIC 

The practice in sixteenth century arithmetics of treating very- 
large numbers under the notation of integers was a prominent 
feature (pp. 34-40). Toms tall, for example, taught the reading 
of integers to five periods and remarked that scarcely ever in 
human experience do larger numbers occur. The treatment of 
large numbers so* early was chiefly due to the topical system 
which required the explanation of reading and writing num- 
bers, as large as the wisest might ever encounter, to be placed 
in the first chapter. Then too, for the purpose of exhibiting 
the idea of periods, a number less than one million was scarcely 
effective. Pestalozzi and his precursors changed all this by 
grading arithmetic, and we might say that large numbers have 
properly disappeared were it not for the gigantic enterprises 
of our century. Let no one endeavor to confine arithmetic 
to thousands when the daily press, in describing the improve- 
ments in our own metropolis, deals with numbers like the fol- 
lowing : " Plans accepted and plans that are certain of accept- 
ance provide for an expenditure of five hundred million dol- 
lars within the next few years/' " Two separate plans for 
the extension of the subway system of New York City rep- 
resenting a cost of two hundred thirty million dollars are now 
under consideration by the rapid-transit commission." " In 
improving the Grand Central Station one million five hundred 
thousand cubic yards of earth will be removed, and thirty 
thousand tons of structural steel be used." "' In all two 
million cubic yards of earth and rock will be carted away before 
the work of building the Pennsylvania Station can be begun." 
Let no one attempt to limit arithmetic to millions when our 
statistical reports contain data like these : " The annual num- 
ber of letters transmitted through our post-ofhees is 20,000,- 
000,000 and of newspapers 12,000,000,000." " The amount 
of deposits in the savings banks of the United States in one 
year was $2,769,839,546." " The total amount of life in- 
surance policies in force is $9,593,846,155, and the value of 
our railroads and equipment is $10,717,752,1 55." 

We cannot deny the necessity for these numbers, but we 
can defer them to the higher grades of the grammar school. 



EDUCATIONAL SIGNIFICANCE 207 

It is only in connection with the larger interests that these 
numibers are met, hence they should be reserved for those pupils 
who are able to appreciate their uses. 

Another feature of the arithmetic of the Renaissance was 
the variety of ways used to perform the processes. E. g., there 
were three methods of subtraction (p. 51), eight methods of 
multiplication (p. 62), and seven methods of division (pp. 
69-72). This excess of variety was characteristic of the en- 
cyclopedic and Latin School writers, and probably was due to 
the tendency to follow Boethius and Hindu-Arabian classics. 
It is quite impossible by reference to sixteenth century arith- 
metic to prove any method to be the preferred one. There 
is precedent for almost every form. Some writers added up- 
ward, and others added downward, (pp. 45-46) ; some added, 
subtracted, and multiplied from: left to right and others from 
right to left (pp. 51, 66) ; some placed the difference above the 
minuend and others below the subtrahend (p. 52). But the 
practical and commercial arithmetics generally presented one 
or two methods, although not always the same ones. The lat- 
ter custom is followed to-day in most cases; but there are 
exceptions, as in the case of subtraction and division. Teach- 
ers are asking : " Shall we teach the plan of taking a unit from 
the next higher order of the subtrahend, or increase the minu- 
end by ten and add one to the next order of the subtrahend ?" 
The following from a New York educational journal shows the 
importance attached to this subject by a superintendent of 
schools : * " Letters received by the committee appointed at 
the Teachers' Institute show that the educators do not agree 
upon any best method, and I feel that it is criminal thus to 
lead the millions of school children into ways that are not the 
best and most practical at an annual expense of millions of 
dollars to the taxpayers." After investigating the subject by 
soliciting opinions from school men and business men his 
conclusion was : " What shall be done when doctors dis- 
agree?" It is safe to say that it is not the author's business 
to determine the only method, but to explain the advantages 
1 Educational Gazette, Vol. 22, No. 5, p. 186. 



2 o8 SIXTEENTH CENTURY ARITHMETIC 

of the leading ones and to present one thoroughly, preferably 
the one personally believed in. This conclusion applies to 
division in the matter of making the divisor a whole number, 
or of writing the quotient above the dividend or at the right 
or under the divisor. 

Something of educational significance attaches itself to the 
old galley method of division (p. 70), for the Italian, or down- 
ward method, to-day recognized to be by far the best of the 
half-dozen ways then in use, scarcely gained a foothold in the 
sixteenth century, although it was known at that time. The 
significance lies in the reason for the persistence of the galley 
method. It possessed the pictorial feature, it aroused curios- 
ity, and it held interest by association with the fascinating 
concept of the ship. 1 There is now a tendency to< abandon 
artificial means such as numbers arranged in squares, triangles, 
and fanciful forms for the purposes of drill on their combin- 
ations. But history tells us that the mind clings to such de- 
vices, and that they are a valuable element in teaching number 
when properly chosen and not used to excess. 

Much is said in discussions of the details of grade work 
about the proper use of language, especially about such words 
as " borrow " and " carry." It would be difficult to> estab- 
lish general usage for these in sixteenth century arithmetic, 
because a translation of a foreign word is seldom unique. But 
there is no mistaking such a statement as Recorde's " keepe in 
mind" (p. 47, n. 1) for carry. 

Probably no teacher of arithmetic would advocate the total 
neglect of short processes, but some prefer to regard them as 
a high polish to be added to the basal attainments of the ele- 
mentary school, while others believe that the shorter forms of 
calculation should be taught in immediate sequence to the cor- 
responding unabridged forms. Sixteenth century custom was 
in accord with the latter plan (p. 75). 

The origin of proofs, or tests of the work of calculation, 

1 The old copy-books contain examples solved by pupils in order to shape 
an attractive form of galley. The same tendency is seen in the odd cro- 
cetta and gelosia methods of multiplication (p. 63). 



EDUCATIONAL SIGNIFICANCE 



'09 



has been explained (pp. 44, 66, 75). It was seen that the 
original reason for their use vanished with the transition from 
the abacus reckoning to figure reckoning and that their use in 
the latter practice decreased through the century. This neg- 
lect increased with the spread of the Hindu system until, with 
the exception of the schools in certain European states, proofs 
of operations practically disappeared from arithmetic. In the 
latter half of the nineteenth century it is safe to say that in 
general the rank and file of the teaching ibody in America had 
no knowledge of the existence of such proofs. The answer 
book, or the teacher's results,, or the answer found by the ma- 
jority of the class under instruction constituted the only court 
of appeal in determining the correctness o'f a solution. But 
the methods of testing operations in the form of casting out 
nines or by reverse processes or by substitution begin to ap- 
pear in recent arithmetics, not because they are an inheritance, 
but because they supply an educational need. For it is now 
generally held that convenient tests are a valuable means of 
developing the self-confidence and independence of the pupil 
in the matter of calculation. Thus, we may not derive our 
present reasons for teaching proofs from sixteenth century 
arithmetic but rather the methods themselves. 

Faith in the Grube plan for teaching the elementary number 
facts is rapidly waning, 1 and sixteenth century arithmetic 
throws some light on this movement. The chief claim for the 
Grube method is the thoroughness secured by successively 
treating each integer in relation to all the integers preceding 
it. But this does not take into account that some combina- 
tions of integers are less useful than others and, therefore, 
should receive less emphasis. This is now recognized as a 
vital defect and one which is not inherited from the masters, 
for, although the early printed arithmetics often contained mul- 
tiplication tables to 36x36 (p. 58), their authors took pains 
to designate the most useful ones. E. g., Tartaglia called the 

1 David Eugene Smith, The Teaching of Elementary Mathematics, p. 
118. McLellan and Dewey, Psychology of Number, pp. 85-92. McMunry, 
Special Method in Arithmetic, p. 46 et seq. 



2IO 



SIXTEENTH CENTURY ARITHMETIC 



tables of 12s, 15s, 20s, 24s, — Venetian, because they were 
of special use in computing with Venetian denominations 
(p. 61). Then the Reckoning Books or practical arith- 
metics of that time devoted little attention to drill on all com- 
binations of numbers in a given field; the products of 10 x 10 
were usually recommended (p. 60, n. 1), but formal lists of all 
these facts were often omitted (pp ; . 65, n. 3, 66, Sluggard's 
Rule). 

We are now passing through a period of neglect of detailed 
explanation of processes with abstract numbers. Writers have 
reached the extreme in this particular, and fuller treatments 
are again appearing in recent books. No doubt this feature 
of arithmetic was exaggerated in the old style books of the 
nineteenth century, but even in themi the practice was not com- 
parable with that of the sixteenth century. The early text- 
books cannot be taken as a guide in this matter, because the 
prominence given to processes by them was largely due to the 
novelty of the Hindu algorism. 

Teachers sometimes fail to distinguish practices which are 
merely conventional from those which should be followed for 
the sake of logic or for educational reasons. This failure is 
sufficiently prevalent to. command attention at state institutes 
for teachers. Among these practices is the one relating to the 
performing of processes in a series. E. g., 2 + 3 X 5 — 6-5-2. 
This has the value 14 and not 9 J or some other value, because 
mathematicians have established this convention for a series of 
operations : the processes oi multiplication and division take 
precedence of addition and subtraction. The convention might 
have been to perform! the operations in order from left to right, 
and some teachers proceed as if that were the case. The 
trouble with these teachers is that they are not aware that it 
is a matter of convention, consequently they do not seek to 
know what the custom is. There is precedent in sixteenth 
century for the present order (p. 99, n. 4), and when such ex- 
pressions occur in text-books the customary method of sim- 
plification should be given. Another question which receives 
some attention in discussions on the teaching of arithmetic is : 



ED UCA TIONAL SIGNIFICANCE 2 1 1 

Which shall be taught first, long- division or short division t 
This is not an arbitrary matter, like the one above referred to, 
but depends upon the educational axiom : Proceed from the 
simple to the complex. In the early printed arithmetics the 
unabridged processes usually preceded the abridged (p. 74). 

The first printed arithmetics also throw some light on the 
teaching of fractions. The fraction from the earliest times 
has been much more difficult to master than the integer. The 
Egyptian tables of unit fractions found in the papyrus of 
Ahmes x exhibit the meagre knowledge of fractions which the 
early civilizations possessed. The Greeks and Romans made 
little progress ; e. g., the Greeks 2 wrote t6' ~\e" Ae'' for Jf , and 
the Romans used a still more clumsy notation. Even the in- 
troduction of the Hindu numerals did not sufficiently simplify 
the work with fractions. The decimal form is the only one in 
which the operations are practically as simple as those with 
integers, but we still have large use for the common frac- 
tions, and their teaching gives rise to several educational ques- 
tions. 

Since the spread of Pestalozzian ideas, the method of pre- 
senting fractions has been through sectioned objects and dia- 
grams. In this plan the unit is made prominent. A plan re- 
cently recommended is that of representing the fraction as a 
ratio, which, in the concrete aspect, depends upon the act of 
measuring, and which emphasizes the collection or group of 
units in place of the absolute 3 unit. A third plan suggested 
by the evolution of our number system is to define the frac- 
tion as an indicated quotient as soon as inexact division is en- 
countered, for the disposition of the remainder gives the logical 
opportunity for introducing the fractional notation. Of 
course, if this were done in primary teaching, concrete illustra- 
tions for the purpose of giving content to the fractional sym- 
bol would not be excluded. It is quite probable that no one of 

1 Eisenlohr, Ein mathematisches Handbuch der alten ^Egypter (Leipzig, 
1877). 

2 Gow, History of Greek Mathematics, pp. 42, 48. 

3 McLellan and Dewey, Psychology of Number, pp. 157-162. 



212 SIXTEENTH CENTURY ARITHMETIC 

these methods is best, and that all three should find place in 
the teaching of fractions. The partition of the single thing, 
the partition of the group, or measured unit, and the quotient 
of two integers all appear to be essential to a general concept 
of the fraction. Furthermore, this is in accord with the treat- 
ments in sixteenth century arithmetics, for Kobel used the first 
plan x and called into service the now illustrious apple (p. 85). 
Writers who combined fractions and denominate numbers used 
the second plan, and others, as Ramus and Raets, used the 
third plan (p. 87). Tonstall suggested an exercise which, 
if used in connection with the last form would assist in clarify- 
ing the pupil's notion of the fraction, namely, the consideration 
of the effect upon the value of the fraction produced by vary- 
ing its terms (p. 89). 

No one seriously questions the logical order of addition, 
subtraction, multiplication, and division in the field of integers 
(the Grube simultaneous plan being a misnomer), but in the 
case of fractions this order may well be violated, even to a 
greater degree than at present. 

The teaching of fractions by sectioning objects leads early 
to such products as 2 X f = I and $ of i = f , not necessarily 
before the presentation of sums like J + f = f, but generally 
before those like ■£ -|- i = A* In other words, teachers of 
primary arithmetic recognize that certain products in the field 
of fractions are more easily found than the sums of fractions 
having different denominators. But writers of text-books, es- 
pecially in their formal treatments, have given little heed to this 
fact, although much was made of this feature during the first 
century of printed books. The Bamberg Arithmetic (1483) 
(p. 181) placed the processes with fractions in the following 
order: multiplication, addition, subtraction, and division. 
Calandri (p. 94) and Paciuolo adopted the same order, and the 
latter explained his position on educational grounds (p. 94). 

The multiplication of a fraction by a fraction gives rise to 
a special difficulty, namely, the explanation of " times " in cases 

1 See also Champenois, p. 81 of this monograph. 



EDUCATIONAL SIGNIFICANCE 



213 



like 5 X 5 = A- For lack of effective means of illustration 
most writers banish the usual symbol, X, and substitute the 
word " of." It would seem: to be a dangerous practice to set 
aside the general notation whenever a difficulty is encountered ; 
and one may reasonably claim Tren- 
chant's plan (1571) to be prefer- 
able, for he showed that, since the 
area of the whole square here 
shown is 1 X 1, or 1, the area of 
the small square should be 5 X 5, or 

A (p. 106). 

There are two methods in com- 
mon use for dividing one fraction 
by another. First, reduce the frac- 
tions to a common denominator 

and divide the numerators. Second, invert the divisor and 
multiply the fractions. The first method is commonly taught 
to beginners, because it is more easily explained; but, since 
the second method is simpler in practice, a transition from the 
first to the second is necessary. It is not the custom of mod- 
ern arithmetics to show the connection between these two 
plans, but sixteenth century arithmetic suggests that we might 
well make the connection in the following way : Divide f by f . 

By the first plan 













1 










Mb 

m 














1 





































1 



3x3 



By the second plan 



4x2 3x3 



3x4 4x3 



4x2 



x l_3_ 



4X2 



3 = 9 
8 



The second plan is the same as the first only the fractions with 
the common denominators are not written. Before writers 
began to invert the divisor they multiplied crosswise (p. 108) 
which is the very thing done in finding the fractions with a 
common denominator, and thus the connection between the 
two methods is made apparent. 

In the problems arising from actual experience, fractions 



2i 4 SIXTEENTH CENTURY ARITHMETIC 

usually occur in connection with denominate numbers, and in 
business problems the aliquot parts of one hundred furnish 
the fractions commonly met. This suggests that writers of 
text-books should place emphasis mainly on fractions whose 
denominators are not greater than one hundred and that little 
attention be paid to fractions whose denominators are prime 
numbers greater than five. This would be in accord with the 
custom of early arithmeticians, for, although they often in- 
dulged in large fractions, the decimal form not having been in- 
vented, they correlated denominate numbers with fractions 
(p. 102) and emphasized the aliquot parts of one hundred 

( P . 81). 

We have already mentioned the importance of denominate 
numbers and the prominent place given to them in the first 
practical arithmetics, but there remain a few points of inter- 
est in relation to the method of presenting them for teach- 
ing purposes. 

The money system of a nation enters so> largely into* the 
data of its arithmetic that this system becomes a controlling 
factor both in the matter of material and of method. The 
money systems of Europe in the sixteenth century were not 
decimal systems, consequently operations involving them were 
comparatively complex. This circumstance coupled with the 
fact that the variety of weights and measures was great 
(p. 83) necessarily led to much adding, subtracting, mul- 
tiplying, and dividing with compound numbers having several 
denominations (p. 84). This practice encouraged by the sur- 
vival of many awkward systems has tended to perpetuate the 
processes with compound numbers to the present time. But 
the increasing popularity of the decimal system and the prac- 
tice of expressing compound numbers in terms of the larger 
units or fractions of the same are sufficient reasons for limit- 
ing the operations with compound numbers to those having 
only two or three denominations. 

Another form of exercise that has not wholly disappeared 
from arithmetic is the reduction from one table of denominate 
numbers to another. E. g.„ find how many pounds Troy there 



EDUCATIONAL SIGNIFICANCE 



215 



are in \j\ lb. avoirdupois. This unquestionably is a type de- 
rived from sixteenth century problems of reduction made 
necessary at that time by the lack of uniformity in the value 
-of current units. An example of this diversity may be seen 
in the following from Riese: * 

10 pounds at Venice = 6 pounds at Nuremberg 

3 centner at Eger =: 4 centner at Nuremberg 

10 pounds at Nuremberg = 11 pounds at Leipzig 

100 pounds at Nuremberg = 128 pounds at Breslau 

7 pounds at Padua = 5 pounds at Venice 

10 pounds at Venice = 6 pounds at Nuremberg 

100 pounds at Nuremberg = 72 pounds at Cologne 

But the day of diversities of this kind having passed so far as 
practical calculation is concerned, it is difficult to find a reason 
for continuing the practice in our arithmetic of reduction from 
one table of denominate numbers to another. 

Another matter in connection with the presentation of de- 
nominate numbers is worthy of notice. The careful grading 
of primary arithmetic and especially the spiral plan (p. 200) 
have tended to distribute the subject-matter of weights and 
measures throughout the whole course in arithmetic. Some 
educators scent danger here, fearing that the subordination of 
so important a subject as denominate numbers may lead to a 
scrappy presentation and unsystematic knowledge. A recent 
remedy is found in the plan of placing a thoroughly organized 
review chapter in grammar school arithmetic. Sixteenth cen- 
tury arithmetic suggests another solution, which consists of 
placing a section on denominate numbers under each process 
with integers and fractions and a final treatment under applica- 
tions. For example, under addition of integers would be 
placed the addition o»f compound numbers, only such numbers 
and reductions being chosen as could be manipulated by pupils 
at that stage; and similarly for other processes. This plan is 
imperfectly shown in the Bamberg Arithmetic which presents 
the topics thus : Addition, Subtraction of Integers ; Addition 
and Subtraction of Denominate Numbers. Mention has been 
made of more detailed examples (p. 94). 

1 Riese, Rechnung auff der Linien und Federn/ (1571 ed.). 



2i6 SIXTEENTH CENTURY ARITHMETIC 

Besides the questions concerning the proper selection of ap- 
plications of arithmetic (pp. 187-199), there are several im- 
portant considerations that relate to their form: of presenta- 
tion. The greatest question of method in this connection is : 
Shall the applications precede or follow the process to be ap- 
plied ? This query seems to imply the absurdity that the effect 
may precede the cause, until one determines which is the cause 
and which the effect. Since every unit of instruction should 
have its aim, and since the consciousness of this aim on the part 
of the pupil is a factor in his interest in the subject, a know- 
ledge of the end for which he studies may well be outlined 
before he is set to acquire the means. In this sense the ap- 
plication may precede the arithmetical process. That is, the 
use to which the process is to be put may be proposed as an 
incentive for learning the process. For ex- 
ample, formal multiplication may be intro 
duced in this way. A bushel of oats weighs 
32 lb. Find the weight of 5 bu. by a shorter 
process than addition. The pupil who 
knows only formal addition uses the first 
process. But when his attention is called 
to the fact that his knowledge of the mul- 
tiplication tables furnishes the sums of the ^ llb l6o j b 
columns he is ready to appreciate the 
second, or shorter form. This is not a new theory for it had 
a large number of adherents in the sixteenth century (pp. 48, 
56, j6). Although arithmetical writers of that time were not 
concerned with primary instruction or method in the modern 
sense, nevertheless they often proposed concrete problems be- 
fore explaining the processes which entered into their solu- 
tion. For example, Champenois introduced the subjects of 
addition and subtraction by proposing military problems 
(p. 128), Baker took commercial problems for his method of 
approach, and Suevus employed facts o'f ancient history 
and fancies quite his own for his center of interest x (pp. 

1 That the attempt to stimulate interest through concrete situations may 



(I) 


(2) 


32 lb. 




32 




32 




32 


32 lb. 


32 


5 


10 


10 


15 


15 



EDUCATIONAL SIGNIFICANCE 217 

129-130. There was one writer, Robert Recorde, who 
employed the motive of utility in a masterly way. His plan 
was the precursor of the modern principle of appealing to the 
pupil's vital interests. It would be difficult to find a better 
type of instruction than his method of approach to* the subject 
of addition by reference to his Oxford pupil's expense account : 

" S. (Scholar) Then wyll I caste the whole charge of one 
monethes comons at Oxford with battelyng also. 

" Master. Go to, let me see how you can doo. 

" S. One wekes comons was 11 d. ob, q. and my battelyng 
that weke was 2 d. q. q. The seconde weekes comens was 
12 d. and my batlyng 3d. The third wekes cbmos 10 d. ob. & 
my batling 1 d. ob, c. The fourth wekes comos 11 d, q, & 
my batling 1 d, ob, c. These 8 sumes wold I adde into one 
whole summe." 

But this method has fallen into complete neglect with the 
passing of the centuries between that time and the present. 
The customary practice is not sufficiently inductive. It pre- 
sents the processes in the fields of integers, of fractions, and of 
denominate numbers as ends in themselves, and treats the ap- 

also lead to artificial means is illustrated toy the ingenuity of Suevus, who 
introduced Roman notation by the story of a famine. 

ON THE GREAT FAMINE IN POLAND AND SILESIA. 

"That the time of the famine may not be concealed, behold cvcvllvm. 
That is, in order that the time of the famine and distress, which long ago 
took place in Poland and Silesia, shall remain concealed from no one, but 
shall be known by all, the year is to be reckoned from the little word 
cvcvllvm, which here means a cap of sorrow." 

m = 1000 

ll = 100 

cc = 200 

wv = 15 



CVCLLVM = 1315 

Sigismund Sueuus, Arithmetica Historica. Die Lobliche Rechenkunst 
(page 64). "Vc lateat nullum tempus famis Ecce cvcvllvm. Das ist: 
Auff das die grosse Thewrung vnd Hungersnot/ die vor zeiten in Polen 
vnd Schlesien gewesen ist/ niemande verborgen bleibe/ sondern von men- 
niglichen wol in acht genommen werde/ so sol man durch das Wortlein 
cvcvllvm, welches hier eine Trawerkappe heist/ die Jahrrechnung machen." 
Page 64. 



218 SIXTEENTH CENTURY ARITHMETIC 

plications as convenient drill work for fixing the methods of 
calculation. Applications, when valued for their own sake, are 
set apart from the processes so as to form an independent 
center of interest. 1 The true course doubtless lies between 
the usage of the present and that of the past. The present 
custom of process worship has led to poverty of ideas, and to 
approach every variety of calculation through a problem in- 
volving its use might lead to* a confusion of ideas. 

Another question of method is this : Is the great prom- 
inence now given to commercial arithmetic in all grades 
justifiable on educational grounds? The present place of 
the subject in instruction is indirectly due to sixteenth 
century teaching; for, when Pes'talozzi founded primary 
arithmetic, he naturally drew his material from the books of 
the Reckoning Schools which had up to that time monop- 
olized instruction in arithmetic. These schools had little 
to offer except commercial arithmetic plus some mensura- 
tion, the two phases of applied arithmetic united by Kobel 
(p. 164). Thus, the arithmetics of the nineteenth century 
became saturated with commercial arithmetic in a narrow 
sense. Money value was emphasized everywhere, " bought 
at " and " sold at " being the usual data. Subjects like part- 
nership were introduced before the pupil had any feeling of 
interest in them, but modern educational research is finding 
more appropriate material for the applications of arithmetic 
in the lower grades, — things which come within the pupil's 
experience and for which he is willing to' study the subject. 
For example, his games,, purchases, and possessions. There 
is a quantitative side also to manual training, domestic art, 
geography, nature study, and drawing. If a boy is making a 
box, a model, or an iron ornament in his manual training work, 
there are related problems about size, amount, and cost of 
materials. If a girl is making an apron, or a book-bag, or 
cooking in domestic art, there are related problems about 
amount and cost of materials. Likewise, the geography reci- 

1 McMurry, Special Method in Arithmetic, p. 113 (New York, 1905). 



EDUCATIONAL SIGNIFICANCE 



219 



tation suggests problems about distances, areas, population, 
and production. Nature study suggests problems about 
weight, time, and motion, and drawing is full of scale meas- 
urements and proportion. If the concrete material of the arith- 
metic hour is drawn in part from recitations in other subjects, 
not only is time saved and review secured, but the ideas of 
arithmetic are enriched by association with varied and vital 
interests. 

Consequently, although sixteenth century arithmetic may 
have led writers to put too much commercial arithmetic into 
the primary course, it set a step in the direction of making 
arithmetic concrete and paved the way for a better and more 
teachable system of subject matter. 

In the early arithmetics mensuration was presented in a 
separate chapter usually placed at the end of the book, which 
place it has since occupied. But we are now coming to a 
more efficient use of this subject in the teaching of arithmetic. 
Fractions are given concreteness by the handling of graduated 
rulers, scale relations are learned by drawing lines and figures, 
decimal fractions are illustrated and made familiar to the stu- 
dent by the use of the meter stick, and so on. 1 That is, the 
facts of measurement and related properties of geometric fig- 
ures are being graded and correlated with number work from 
the fourth year to the eighth. This plan has the further ad- 
vantage of preparing for the geometry of the high school, and 
of offering the opportunity for practical field mathematics, 
through determining heights, distances, and areas. 

There is a class of problems that may be called factitious or 
artificial. For example, 100 potatoes are placed in a row at 
intervals of 10 yd. A basket is placed at one end of the row, 
how long will it take a person, who can walk 20 yd. a minute, 
to bring all the potatoes to> the basket ? In the sixteenth cen- 
tury such problems served as applications to the progressions 
(p. no), and similar ones to proportion and evolution. The 

1 F. T. Jones, in School Science and Mathematics (Chicago), Vol. 5, 
No. 6, p. 408 (iox>5)- 

2 W. T. Campbell, Observational Geometry, Boston (1901). 



220 SIXTEENTH CENTURY ARITHMETIC 

early writers understood the function of these exercises to be 
drill in processes which had no real applications, but which 
were valuable because of their importance in other branches 
of mathematics. But now, since the processes themselves have 
been promoted to more advanced mathematics, there is no good 
reason for retaining these artificial problems in elementary 
arithmetic. 

Although there is no> single rule that will solve all of the 
problems of arithmetic, there are two general methods which 
have wide application, unitary analysis and the equation. All 
grades of problems from "If one orange costs 3 cents what 
would 5 oranges cost?" to " If 5 men working 8 hours a day 
can dig a trench 3 ft. wide, 12 ft. deep and 300 ft. long in 
10 days., how many men will it take working 10 hours a day 
to dig a trench 3 J ft. wide, 18 ft. deep, and 450 ft. long?" 
may be solved by either method. Unitary analysis, although 
known in the sixteenth century (p. 137) was commonly modi- 
fied into the Rule of Three (p. 132). This rule, a great 
favorite for centuries, is still preserved in the subject of Pro- 
portion (p. 134), a method now yielding to the equation. The 
last method, too, should not be thought of as a modern in- 
vention, but rather as a modern form of the sixteenth cen- 
tury Rule of Three. The solution of the following example 
from Thierfeldern by the Rule of Three shows a marked 
similarity to the present method by the simple equation : " If 
18 florins minus 85 groschens are equal to> 25 florins minus 
2^2 groschens, how many groschens are there in 1 florin?" * 

1 Caspar Thierfeldern, Arithmetica Oder Rechenbuch Auff den Linien 
vnd Ziffern/ (1587). 

" Item/ 18 ft. weniger 85 gr. machen gleich so vil gr. als 25 ft. -f- 232 gr. 
wie vil hat 1 ft. groschen? Facit 21 gr." 

" In disem beyden Exempeln/ addir das Minus/ und subtrahir das Plus/ 
wie hie." Page no. 



18 ft. 


-5- 85 gr. gleich 25 ft. 


-r- 232 gr. 
85 gr. 


18 ft. 


gleich 25 ft. 


-J- 147 gr. 


18 ft. 


~- 147 gr. gleich 25 ft. 
18 






147 gr. gleich 7 ft. 





EDUCATIONAL SIGNIFICANCE 221 

These solutions I and II depend upon the laws of transpo- 
sition and are the same except that the unknown quantity, x, 
is missing from the first. But the ease with which the second 
form can be followed shows the advantage oi the modern plan 
with x. Thus, although changed to a more convenient form 
by improved symbollism, the great method of the sixteenth 
century will become the favorite one of the twentieth. 

Undoubtedly, one of the chief reasons why the equation did 
not take its place in arithmetic much earlier is the prominence 
given to the Rule of False Position (p. 153). The use of 
this method of approximation having been introduced from 
algebra delayed the more definite process of the equation. 

In considering the form of presentation best suited to the 
more advanced applications of arithmetic one must take into 
account that arithmetic is an important factor in interpreting 
the world about us. " It is a standpoint from which the bet- 
ter to see through and around a great many important topics. 

Without the illumination from mathematics a great many 
important facts and bodies of knowledge in geography, his- 
tory, natural science, and practical life remain hazy and not 
clearly intelligible." 1 Thus, the concrete material from which 
the processes of arithmetic may be taught should do more than 
furnish the center of interest. It should furnish types of 
quantitative experience of use in appreciating the larger inter- 
ests of life. " It is now thought proper to take a class to a 
saw-mill, a stone quarry, a cotton factory, or a foundry as to 
a laboratory or a recitation room. The industries of the 

7 a. 147 gr. 1 n. 

facit 21 

II. 1. Let x = the number of groschens in 1 florin 

2. Then i8x — 85 = 25X — 232 

3. Therefore, i8x = 25X — 147 

4. " i8x + 147 = 25X 
5- " 147 = 7x 

6. 7x = 147, and x = 21. 

1 McMurry, Special Method in Arithmetic (New York, 1905), pp. 113- 
114. 



22 2 SIXTEENTH CENTURY ARITHMETIC 

neighborhood become standards by which the world's work of 
various sorts is estimated and judged." * 

Arithmeticians have been very slow to grasp this inter- 
pretative function of arithmetic while writers of text-books in 
other subjects have made marked progress in this direction. 
Geography, for example, no longer consists merely of unre- 
lated facts about the political divisions and physical features 
of the earth, but treats of the influence of heat, moisture, soil, 
climate, and other factors on resources and industries together 
with the significance of the latter in determining national con- 
ditions and peculiarities. One cause which has helped to keep 
arithmetic in this backward condition is the disciplinary ideal 
inherited from the Latin Schools of the sixteenth century 
(pp. 174-178). This ideal has led writers and teachers to 
look upon the applications of arithmetic as an instrument of 
discipline to the exclusion o>f its larger function of informa- 
tion giving. For example, the chapter on percentage was 
given over by some to the consideration of the mechanics of 
its nine cases : 

Base X rate = percentage Percentage -s- rate =base 

Base X (1 + rate) = amount Amount -4- (1 -f- rate) = base 

Base X (1 — rate) = difference Difference -5- (1 — rate) = base 

Percentage ■+■ base = rate Amount -s- base = 1 -f rate 

Difference ■+■ base = 1 — rate 

instead of to its use in answering the quantitative ques- 
tions of commerce and of the crafts and sciences. Another 
cause is the recent reaction against the topical plan. Six- 
teenth century arithmetic presented all its subject-matter both 
theoretic and applied arranged by topics (pp. 29-170), which 
remained the standard form until the closing decade of 
the nineteenth century when a school of writers favoring ex- 
treme gradation reduced the applications of arithmetic to a 
mass of unrelated questions. It is now necessary to reorgan- 
ize the problems of arithmetic into groups and vitalize them 
so that they may shed light upon the various topics which 
school subjects should illuminate. 2 

1 Dutton, School Supervision, p. 208 (New York, 1905). 

2 Smith and McMurry, Mathematics in the Elementary School (New 
York, 1003), PP- 58, 59- 



EDUCATIONAL SIGNIFICANCE 



223 



Mode 

In addition to the questions which relate to the selection of 
subject-matter and to its organization for teaching purposes, 
there are questions concerning the mode of class-room in- 
struction. By mode is here meant the form of class exercise. 

Although, directly, sixteenth century arithmetic throws little 
light upon this department of teaching, indirectly it has im- 
portant bearings. For, among the modes in use to a greater 
or less extent at the present time, the early printed arithmetics 
touch in a significant way the heuristic, the individual, the lec- 
ture, the recitation, and the laboratory modes. The bearing 
is an indirect one, because it is largely through the method of 
the subject-matter that one must determine the mode of teach- 
ing at that time. This plan of investigation, which would, 
in general, be misleading, is quite safe in the present instance, 
because the authors of many of the most significant arithmetics 
were prominent teachers; and besides formulating the subject- 
matter in harmony with their favorite mode, they often named 
in the prefaces or introductions to the works the kind of teach- 
ing for which their arithmetics were adapted (pp. 178, 186). 

The heuristic mode finds its prototype in the catechetical 
books (p. 202). There can be no mistaking the form which 
teaching assumed that drew its material from these books. 
The purely catechetical mode, of course, was not heuristic, 
but such a treatment as Record e's (p. 202) contains the de- 
veloping process, the real unfolding of new ideas by means of 
suggestion and skillful questioning. Perhaps the most im- 
portant lesson to be drawn from this is that the heuristic mode 
of teaching should not be carried to an extreme, that is, should 
not reduce all instruction to a system of interrogations lest it 
suffer the fate of the purely catechetical form (p. 203). A 
lesson of secondary importance concerns the function of oral 
arithmetic. If the historical precedent be followed, we may 
infer from Fischer, Suevus, and Recorde that the oral work of 
the recitation should be given more to the development of new 
ideas than to drill upon those already taught. 

Although on account of the vastness of public education in 



224 SIXTEENTH CENTURY ARITHMETIC 

this country, mass instruction will be inevitable for a long time 
to come, much is said about the advantages of individual in- 
struction, and many devices are being employed to 1 obviate the 
defects of the class system. Among these is the laboratory 
mode discussed on page 225. The common plan of teaching in 
the sixteenth century was the individual one (p. 180), and 
many arithmetics were written in conformity with this practice. 
It was often stated in the prefaces of these books that they 
were also adapted to> self-instruction. This was true, for ex- 
ample, of the Bamberg Arithmetic, of Kobel's Zwey Rechen- 
buchlin, and of Recorde's The Ground of Artes. The char- 
acteristic feature of these books, excluding an occasional 
work like Recorde's, was the direct presentation of processes. 
The formal rule, the "Thue ihm also" o'f Riese, was the sign- 
board over every new path. Thus, history suggests that the 
present danger to arithmetic in individual instruction may 
consist in this, that the teacher who attempts to instruct even 
a small class on this plan is liable to fall into the error of dic- 
tating method instead of directing it. 

The practice of merely assigning lessons to be learned out- 
side of school hours and recited at the next meeting of the 
class, commonly called the recitation mode, is an outgrowth of 
modern machine teaching. There is no* evidence that arith- 
metic was taught in this fashion in the sixteenth century; on 
the contrary, the contents of the text-books and the use of the 
separate problem book x indicate that the time spent in school 
was given to instruction, and that the work assigned was in 
the form of applications. It is safe to say that sixteenth 
century teaching of arithmetic with all its faults was better 
than the so-called recitation mode. 

There is still an occasional educator who believes in the 
lecture mode of instruction, who' holds that text-books have 
disada vantages which outweigh their usefulness; but, since 
there is no likelihood of this mode's becoming common in our 
elementary schools,, it may be passed over briefly. In the six- 

1 As Rudolff's Exempel Buchlin (p. 182, this monograph). 



EDUCATIONAL SIGNIFICANCE 225 

teenth century arithmetic was taught in the universities. In 
fact, several authors, as Ramus, Widman, and Apianus were 
university professors. It is undoubtedly true that arithmetic, 
like other university subjects, was often taught by lecture, but 
there is nothing in the early teaching of arithmetic that sug- 
gests the advisability of using the lecture mode in the modern 
elementary school. 

Among the ways most discussed at present for presenting 
subject matter in the class room is one known as the laboratory 
mode. The characteristic feature of this mode consists in the 
teacher directing the pupil in discovering and verifying im- 
portant truths by the aid of some form of laboratory equip- 
ment. In mathematics the equipment consists of books, maps, 
charts, blanks, legal forms, drawing materials, physical appar- 
atus, and other materials suited to the subject in hand. So 
far, the experiments in the use of this plan have practically 
been confined to secondary schools., but there is room for at 
least an adaptation of it in the elementary school. The his- 
tory of the formative period of commercial arithmetic shows 
that its teaching began with this mode (p. 178 et seq.). The 
precursor of the Reckoning School was a system of apprentice- 
ship. A knowledge of arithmetic was obtained by working 
in the ledgers of the counting-house, by keeping the public 
records of the municipality, by listing, weighing, and measur- 
ing in the warehouse and by serving the surveyor in the field. 
The modern business college mathematics in which nominal 
sales, shipping, and banking departments lend a semblance of 
reality to the work are in harmony with the historical de- 
velopment of business arithmetic. And one may reasonably 
ask, in view of the present movement toward the vitalization 
of arithmetic, toward its application to manual and domestic 
arts, and toward the recognition of its interpretative function in 
the economic questions of our people, if the spirit of the labor- 
atory mode may not be a needed element in teaching element- 
ary arithmetic in our public schools ? 



226 SIXTEENTH CENTURY ARITHMETIC 

SUMMARY 
From the foregoing we may conclude that the teaching 
of arithmetic in the sixteenth century supports the following 
general theses : 

Concerning Subject-Matter 

i . Rapid commercial and industrial development has a vital- 
izing effect upon the subject-matter of arithmetic. It tends 
to enrich its problems; it encourages improved methods of 
calculation and conditions the selection of denominate num- 
bers. Commercial development had this controlling tendency 
in the sixteenth century, and it is exerting the same influence 
in the twentieth century. 

2. The disciplinary function of arithmetic reaches its great- 
est efficiency through the uses of number rather than through 
the properties of numbers, a principle not generally recognized 
in the Latin School books. The culture ideal has always led 
to the selection of subject-matter having a many-sided inter- 
est, and the propaedeutics of arithmetic at present, as in the 
past, require the retention of certain theoretic matter. 

3. Traditional custom' is not a safe guide in the selection of 
subject-matter, since it tends to perpetuate obsolete material. 
Present commercial, industrial, and educational needs are the 
true basis of such selection. 

4. The two sixteenth century schemes of arranging subject- 
matter, namely, by kinds of numbers and by kinds of processes 
have proved failures in modern graded curricula. Present 
needs can be met only by a combination of the two plans. 

Concerning Method 

1. The psychologic method produces ideal modern books. 
The arithmetic of the Renaissance furnishes excellent speci- 
mens of the synthetic and analytic methods, and marks the 
birth of the psychologic. The last method is eclectic, taking 
the best features from the other two, and, besides, pursues the 
path of least resistance in accord with modern educational 
principles. 



EDUCATIONAL SIGNIFICANCE 227 

The following details of method find support in the first 
century of printed arithmetics : 

1. Artificial means for making number work interesting 
should not be abandoned. 

2. Number combinations are not equally important. 

3. Unabridged processes should precede abridged ones. 

4. Methods of testing the work of calculation should be 
given. 

5. Definitions are desirable in arithmetic. In cases of pro- 
cesses they should tell how the operations are performed. They 
should also admit of easy extension to cover the processes 
when new fields of numbers are entered. Sixteenth century 
definitions are superior to most modern ones in this respect. 

6. In regard to notation we may learn from the results of 
conservatism in the past how necessary it is to welcome im- 
proved symbolism. 

7. Our books should explain the processes by the generally 
preferred methods. The best text-books among the early 
arithmetics did this. Encyclopedic works, now as well as 
then, are valuable only for reference. 

8. The partitive idea, the measuring idea, and the ratio 1 idea 
are necessary to the concept of the fraction. All of these 
ideas were recognized in sixteenth century arithmetic. 

9. In the formal treatment of fractions the logical order, 
addition, subtraction, multiplication, and division should be 
replaced by the psychological order, multiplication, addition, 
subtraction, and division. 

10. Fractions should be correlated with denominate num- 
bers, 

11. The presentation of denominate numbers should be dis- 
tributed under the processes with integers and fractions. In 
compound numbers only those of two* or three denominations 
require emphasis, and reductions from one table to another 
are no longer necessary. 

12. The applications of arithmetic may be proposed as in- 
centives for learning the processes. They should be classified 
in accordance with the aims and motives of the different 



228 SIXTEENTH CENTURY ARITHMETIC 

periods of school instruction and should perform an inter- 
pretative function. 

13. The simple equation is an improved form of sixteenth 
century analysis and promises to become the favorite method 
for solving the problems of arithmetic. 

Concerning Mode 

1. The heuristic mode as used in the sixteenth century sug- 
gests that oral work should be used chiefly to develop new 
ideas. 

2. The individual mode is apt to result in dogmatic in- 
struction. 

3. The recitation mode finds no precedent in early teaching. 

4. The lecture mode has no place in elementary arithmetic. 

5. Renaissance arithmetic suggests the use of the laboratory 
mode. 

Thus, we conclude that the history of the formative period 
of arithmetic supplements in many ways the conclusions of 
educational theory in regard to the subject-matter, method, 
and mode of modern arithmetic. 



INDEX 



Abacist, 53 

Abacus, 25, 46, 74, 209 

Abstract Problems, 56, 192 

Addends, 51 

Addition, 26, 35, 201 

of integers, 41, 61, 175 
of fractions, 98 
Ahmes, 211 
Alcuin, 160 
Algorism, 27, 61, 168 
Al Khowarazmi, 36 
Aliquot parts, 81, 82 
Alligation, 132, 151, 157, 198 
Amenities, 194 
Andres, 28 
Apianus (Bienewitz), 27, 28, 150, 

181, 182, 225 
Applications, 216, 218, 220, 222, 227 
Applied Arithmetic, 97, 127, 129, 

131, 169, 176, 198 
Approximations, 66 
Arabs, 36 
Archimedes, 177 
Area, 79 
Aristotle, 29 
Arithmetic, 35 
Articles, 38 
Aryabhatta, 132 
Ascham, 171 
Assize of Bread, 132, 157 
Associative Law, 47, 100 
Aventinus, 28 
Bachet de Mezeriac, 159 
Baker, 23, 30, 48, 53, 69, 74, 75, 81, 

97, 99, 103, 108, 109, 113, 121, 138, 

142, 151, 183, 186, 188, 216 
Bamberg Arithmetic, 27, 76, 212, 

215, 224 
Banking, 146 
Barter, 132, 150 
Bhaskara, 36 
Binomial Coefficients, 113 
Biordi, 158 
Bocher, 191, 192 
Boethius, 30, 35, 187, 207 
Bookkeeping, 183 
Borgi, 37, 39, 52, 57, 61,67,94,110, 

133 
Borgo. See Paciuolo. 
Bosanquet, 90 



Brahmagupta, 149 

Brooks, 27, 53 

Business Applications, 186 

Buteo, 31, 119, 134 

Calandri, 67, 69, 71, 75, 94, 104, 131, 

137, 160, 169, 183, 187, 212 
Calculation, 66, 78, 85, 86 
Campbell, 219 

Cantor, 27, 29, 36, 65, 76, 123, 148 
Cardanus, 37, 42, 84, 100, 154, 173, 

186, 196, 199, 202 
Casting Accounts, 23 
Casting out Nines, 45, 52, 74 
Casting out Sevens, 66, 74 
Cataldi, 39 

Cataneo, Girolamo, 78, 79 
Cataneo, Pietro, 60, 150, 164 
Catechetical Arithmetic, 202 
Chain Rule, 132, 148 
Champenois, 30, 45, 48, 49, 56, 69, 

73, 75, 86, 104, 105, 106, 114, 116, 

121, 128, 216 
Chiarini, 80, 158 
Chuquet, 39 
Ciacchi, 101, 150 
Cicero, 129, 172 
Circular Numbers, 33 
Cirvelo, 39, 47, 58, 65, 84, 90, 91, 126 
Classification, 32-34, 187 
Clavius, 47, 173 
Clichtoveus, 35 
Commercial Arithmetic, 35, 83, 179, 

187, 207, 219 

Commercial Problems, 57, 169, 178, 

184, 191 
Commission, 199 
Common Denominator, 97, 98, 101, 

108 
Compayre, 171 

Complementary Multiplication, 65 
Complex Fractions, 197 
Composite Numbers, 32 
Conant, 146 
Concrete Problems, 46, 48, 56, 68, 

76, 216 
Correlation, 194 
Counters, 24, 46 
Courses of Study: 

in Latin Schools, 172 
in Reckoning Schools, 180 
229 



230 



INDEX 



Cube Root, 125, 197 
Cubic Numbers, 33 
Culture Value, 178 
Cunningham, 141, 150, 157 
Custom House, 83 
Decimals, 66, 95, 197, 198, 214 
Dedekind, 32 
Defective Numbers, 33 
Definitions, 191, 204, 227 

of numbers, 29 

of processes, 35 

of fractions, 85 
De Morgan, 23, 34, 36 
De Moya, 183 
De Muris, 127 

Denominate Numbers, 49, 56, 60, 
77-85, 97, 135, 184, 186, 190, 198, 
214, 215, 217, 226 
Dewey, 211 
Diagrams, 106, 124 
Digits, 38 
Diophantus, 177 
Di Pasi, 83, 148 
Discount, 169, 198 
Division, 27, 36, 201 

of integers, 69-76 

downward method, 70, 71 

twelve cases, 73 

of fractions, 107 
Doubling, 36, 47, 76, 109 
Duplatio. See Doubling. 
Duties, 169 
Dutton, 222 
Eisenlohr, 211 
El Hassar, 36 

Equation of Payments, 132, 145 
Equations, 55 

Euclidean Method of G. C. D., 96 
Evolution, 121-127 
Exchange, 24, 132, 146, 158, 199 

Bills of, 146 
Factor Reckoning, 132, 142 
Finaeus, 75, 119, 157, 165 
Finger Reckoning, 28 
Fischer, 172, 174, 180, 223 
Fractions, 85-110, 227 

Classes of, 90-92 

Definitions of, 85-90 

Order of processes in, 93-95 

Reduction of, 95-98 

Teaching of, 212-214 
Freidlein, 24 

Galley Method, 69, 70, 208 
Gauge, 145, 165, 184 
Gemma Frisius, 30, 36, 37, 59, 69, 
76, 77, 86, 94, 103, 132, 138, 153, 
173, 176, 177 
Gerhardt, 51 

Ghaligai, 94, 122, 150, 159 
Gio, 88 



Girard, 174 

Golden Rule, 132 

Gow, 34, 211 

Grammateus. See Schreiber. 

Graphical Methods, 106, 213 

Greatest Common Divisor, 95,96, 197 

Grube, 209, 212 

Halving, 36, 76 

Hanseatic League, 178, 179 

Heer, 133, 150, 180 

Henry, 24 

Herbartian Preparation, 139 

Hindu Numerals, Symbols, 24-26, 

37, 41, 186, 209 
Hispalensis, 90 
Incommensurables, 35 
Integers, 197 
Interest, 132 
Annual, 198 
Compound, 145, 199 
Simple, 143 
Tables of, 144 
International System, 80 
Inverse Operations, 74, 199 
Inverse Rule of Three, 132, 138 
Involution, 121-127 
Jacob, 45, 132, 152, 158, 181, 183 
Jacoba, 36 

Jean, 82, 83, 144, 168 
Jewish Profit, 145 
Jones, 219 

Jordanus, 35, 76, 119 
Kastner, 26 

Kettensatz. See Chain Rule. 
Knott, 26, 27 
Kobel, 37, 46, 48, 53, 60, 74, 76,81, 

85, 91, 107, 121, 164, 181, 184, 186, 

212, 218, 224 
Kuckuck, 23, 24, 27 
Lamy, 171 
La Roche, 39 
Latin Schools, 170, 178, 179, 190, 

204, 222 
Least Common Multiple, 197 
Legendre, 34 
Leonardo, 76, 149, 150 
Leslie, 27, 29 
Licht, 26 
Lilivati, 36, 51 
Linear Numbers, 32 
Logarithms, 168 
Luther, 171, 173 
Magic Squares, 195 
Masudi, 159 
Maurolycus, 30, 35, 177 
McLellan, 211 
McMurry, C, 218, 221 
McMurry, F., 222 
Mediatio. See Halving. 
Melancthon, 171 



INDEX 



231 



Mensuration, 163, 184, 218 
Method, 185, 201 

Analytic, 226 

Psychologic, 226 

Synthetic, 226 
Million, 38, 39 

Mint and Mintage, 132, 147, 151 
Mixed Numbers, 101, 103, 105 
Mode, 185, 223, 228 

Heuristic, 223, 228 

Individual, 224, 228 

Laboratory, 225, 228 

Lecture, 224, 228 

Recitation, 224, 228 
Monroe, 30 
Moya, 29 
Muller, 54 
Multiplication, 27, 35, 201 

of integers, 57-69 

Eight methods of, 62-65 

effractions, 104 
Nasmith, 157 
Neander, 171 
Nicomachus, 30, 35, 187 
Notation, 37, 205, 206 
Noviomagus, 28, 29, 42, 98, in 
Numeration, 37, 129 
One-to-one Correspondence, 32 
Onofrio, 154 
Operations, 76, 77, 191 
Order of Processes, 212 
Orders, 38 
Ortega, 141, 183 
Overland Reckoning, 132, 157 
Paciuolo, 28, 29, 32, 39, 45, 61, 64, 
71, 72, 94, 126, 146, 169, 186, 187, 
195, 202 
Partial Payments, 198 
Partnership, 132, 139, 140, 198 
Peacock, 23 
Peirce, 191 
Percentage, 158, 212 
Per cent Sign, 158 
Perfect Numbers, 33 
Pestalozzi, 206, 218 
Piscator. See Fischer. 
Place Value, 38 
Planudes, 69, 90 
Plato, 30, 178 
Polygonal Numbers, 30 
Powers, 121 

Practical Arithmetic, 24, 80, 117, 128 
Primary Arithmetic, 218 
Processes, 27, 73, 175, 184, 207, 226, 
227 

with fractions, 93 

with integers, 36, 83 
Profit and Loss, 132, 198 
Progressions, 110-116, 186, 219 
Proofs, or Tests, 44-46, 52, 66, 74, 
109, in, 208 



Proportion, 1 17-120, 132, 134-135, 

i85, 220 
Ptolemy, 177 
Puzzles, 159-163, 186 
Quadrans, or Quadrant, 165 
Quadrivium, 190 
Rabelais, 171 
Raets, 38, 44, 80, 87, 97, 100, 143, 

183, 187, 212 
Ramsey, 141 
Ramus, 36, 48, 50, 52, 59, 89, 95, 58, 

120, 134, 172, 173, 174, 176, 186, 

202, 225 
Rashtrakuta, 31 
Ratio, 117-120 

Reckoning. See Abacus, Finger, 

Hindu, Counters. 
Reckoning Book, 164, 174, 181, 

187-9, 210 
Reckoning Master, 178, 179, 180, 184 
Reckoning Schools, 178, 179, 180, 

218, 225 
Recorde, 22, 28, 38, 39-41, 47, 49, 

183, 188, 202, 208, 217, 223, 224 
Regula del chatain, 150 
Regula Fusti, 132, 152 
Reichelstain, 184 

Renaissance, 23, 30, 170, 190, 207, 226 
Rents, 132, 157 
Riccardi, 23 
Riese, 23, 60, 65, 67, 77, 84, 97, 99, 

no, 112, 136, 140, 142, 145, 146, 

149, 153, 169, 173, 181, 182, 187, 

203, 204, 215, 224 
Rodet, 132 

Roman Symbols, 24, 27, 28, 37, 91, 

205 
Roots, 37, in, 113, 121-127, 186 
Rouse, 172 

Rudolff, 26, 57, 60, 84, 94, 96, 99, 
101, 108, 143, 148, 150, 151, 169, 
181, 182, 184, 187, 196, 224 
Rule of Drinks. See Virgin's Rule. 

of False Position, 132, 153,221 

of Three, 109, 131, 132, I37-I39, 
142, 149, 220 
Rule of Two, 138 
Rules, Minor, 156 

in verse, 184 
Salaries of Servants, 132, 157 
Sayce, 90 
Savonne, 183 

Schools of Teaching Orders, 170 
Schreiber (Grammateus), 183 
Scratch Method, 69, 122 
Seeley, 171 
Series, no, in 

Arithmetic, no 

Geometric, no 

Harmonic, no 

Sfortunati, 142 



232 



INDEX 



Short Division, 72, 74 

Short Methods, 46, 47, 67, 75 

Sluggard's Rule, 65, 66 

Smith, D. E., 23, 158, 194, 209 

Solid Numbers, 33 

Species, 36, 77, no 

Square Numbers, 33 

Steinschneider, 153 

Sterner, 26, 29 

Stevinus (Stevin), 30, 96, 120, 172 

Stifel, 23, 150, 172, 173, 174 

Stocks, 199 

Stoy, 29 

Sturm, 171, 184 

Subject-Matter, 185, 186-201, 226 

Subtraction, 26, 3s, 45 
of fractions, 102 
of integers, 49, 53, 55, 61 

Suevus, 37, 129, 130, 131, 139. 217, 
223 

Superficial Numbers, 33 

Superfluous Numbers, 33 

Suter, 36, 51 

Syllogism, 191 

Symbolism, S3~55 

Tables, 41, 44, 50, 56, 57, 59, 82, 92, 
144 

Tagliente, 28, 75 

Tare, 169 

Tartaglia, 35- 41, 44, 45, 50, 53, 58, 
61, 65, 66, 70, 72, 73, 75, 87, 88, 
93, 99, 119, 121, 146, 147, 154, 168, 
169, 186, 187, 196, 202, 209 

Taxes, 169 

Terence, 172 

Theoretic Arithmetic, 24, 128, 196 

Thierfeldern, 55, 107, 108, 113, 121, 
151, 153, 170, 220 



Toilet Reckoning, 27 

Tonstall, 35, 39, 42, 49, 50, 58, 65, 

73, 89, 100, 103, in, 117, 118, 124. 
125. 141, 143, 154, 175, 206, 212 

Trenchant, 35, 36, 46, 49, 53, 56", &7> 

74, 87, 98, 102, 106, 122, 123, 124, 
128, 142, 146, 158, 183, 213 

Treutlein, 23, 27, 69, 90 1 

Treviso Arithmetic, 24, 39, 138, 183, 

187 
Triangular Numbers, 33 
Trotzendorf, 170 
Unger, 37, 39, 53, 146", 149, 170, 171, 

172, 179, 180 
Unicorn, 31, 45, 53, 65, 72, 88, 117, 

121, 186, 195, 202 
Unitary Analysis, 137, 138, 169, 220 
Unity, 29, 30 
Usury, 141 
Van Ceulen, 120, 183 
Van der Scheure, 55, 92, 105, 109, 

no, in, 143, 145, 156, 186, 187 
Villicus, 27, 29, 150 
Vincentino, 119 

Virgin's Rule, RegulaCecis, 132, 153 
Visirbiich, 164 
Voyage, 132 
Wagner, 181, 187 

Weights and Measures, 80, 85, 120 
Weissenborn, 30, 65 
Welsch Practice, 132, 135-137 
Wencelaus, 54, 72, 104, 158, 183 
Widman, 27, 28, 38, 54, 77, 126, 127, 

149, 150, 154, 162, 181, 225 
Willichius, 202 
Woepcke, 24 
Wolf, 184 
Zero, 29, 31, 42 



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